^ 


UNIVEfiSITY   OF  CALIFORNIA 

LIBRARY 

OF  THE 

DEPARTMENT   OF   PHYSICS 


Received. 


OCT  1  2 1908 


Accessions  No. 


il2^ 


Book  No /. 


Driao* 


>t;^ 


I 


1 


LOWER  DlViSIOI^ 


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THE   ELEMENTS    OF    MECHANICS 


^^^ 


INTERNATIONAL   PROTOTYPE    KILOGRAM    No.  20. 

A0REE8  WITH   THE   STANDARD   KILOGRAM   TO   ABOUT   ONE   PART   IN    A   HUNDRED   MILLION. 


THE 

ELEMENTS  OF  MECHANICS 


A  TEXT-BOOK   FOR  COLLEGES 
AND    TECHNICAL    SCHOOLS 


BY 

W.  S.  FRANKLIN-T^tSf^gARRY   MACNUTT 


THE    MACMILLAN   COMPANY 

LONDON  :   MACMILLAN  &  CO.,  Ltd. 
•       1907 

All  rights  reserved 


PHVS/CS  DEPi; 


PHYSICS    DEPT. 

Copyright,  1907 
By  the  MACMILLAN   COMPANY 


Set  up,  electrotyped  and  published  June,  1907 
November,  1907 


*      •  •  •     •  a 


•  •  »•• » 


f>RES8  OF 

The  New  Era  Printing  Company 
Lancaster,  pa. 


PREFACE. 

In  attempting  to  explain  the  object  which  we  have  had  in  view 
in  preparing  this  elementary  treatise  on  Mechanics,  we  will 
assume  that  those  to  whom  this  preface  is  addressed  have  read 
our  introductory  chapter  and,  particularly,  the  section  entitled 
The  Science  of  Physics. 

We  consider  that  the  most  important  function  of  the  teacher  of 
physics  is  to  build  the  logical  and  mechanical  structure  of  the 
science ;  the  logical  structure  mainly  by  lecture  and  recitation 
work,  including  a  great  deal  of  practice  by  the  student  in 
numerical  calculation,  and  the  mechanical  structure  of  the  science 
mainly  by  laboratory  work.  We  believe  that  these  two  aspects 
of  physics  study  should  run  along  together.  This  text,  however, 
is  intended  as  a  basis  for  the  work  of  the  class  room. 

One  difficulty  in  the  teaching  of  physics  is  that  the  native  sense 
of  most  men  is  incapable,  without  stimulation,  of  supplying  the 
materials  upon  which  the  logical  structure  of  the  science  is 
intended  to  operate.  Those  modern  studies  in  psychology  which 
have  brought  to  light  the  so-called  marginal  regions  of  the  mind 
have  surely  an  important  bearing  upon  this  question.  All  ele- 
mental knowledge  such  as  the  knowing  how  to  throw  a  ball,  how 
to  ride  a  bicycle,  how  to  swim,  or  how  to  use  a  tool,  seems  to  be 
locked  in  these  marginal  regions  as  a  very  substantial  but  very 
highly  specialized  kind  of  intuition,  and  the  problem  of  teaching 
elementary  mechanics  is  perhaps  the  problem  of  how,  by  sug- 
gestion or  otherwise,  to  drag  this  material  into  the  field  of  con- 
sciousness where  it  may  be  transformed  into  a  generalized  logical 
structure  having  traffic  relations  with  every  department  of  the 
mind.  A  formal  and  abstract  treatment  of  the  principles  of  ele- 
mentary mechanics  tends  more  than  anything  else  to  inhibit  the 
influx  of  this  elemental  knowledge  from  the  marginal  regions  of 

673202 


VI  PREFACE. 

the  mind  into  the  field  of  consciousness,  and  results  in  the  build- 
ing up  of  a  theoretical  structure,  which  can  have  no  effectual 
traffic  with  any  mental  field  beyond  its  own  narrow  boundaries. 
Such  a  state  of  mind  is  nothing  but  a  kind  of  idiocy,  and  to  call 
it  a  knowledge  of  mechanics  is  silly  scholasticism. 

Another  difficulty  is  that  the  human  mind,  intuitively  habituated 
as  it  is  to  consider  the  immediate  practical  affairs  of  life,  is  hardly 
to  be  turned  to  that  minute  consideration  of  apparently  insignifi- 
cant details  which  is  so  necessary  in  the  scientific  handling  even 
of  the  most  distinctly  practical  problems. 

A  third  difficulty,  which  indeed  runs  through  the  entire  front- 
of-progress  of  the  human  understanding,  is  that  mind-stuff,  which 
has  been  developed  as  correlative  to  certain  aspects  of  our 
ancestral  environment,  must  be  rehabilitated  in  entirely  new  rela- 
tions in  fitting  a  man  for  the  conditions  of  civilized  life.  Every 
teacher  knows  how  much  coercion  is  required  for  so  little  of  this 
rehabilitation  ;  but  the  bare  possibility  of  the  process  is  a  remark- 
able fact,  and  that  it  is  possible  to  the  extent  of  bringing  a  New- 
ton or  a  Pasteur  out  of  a  hunting  and  fishing  ancestry  is  indeed 
wonderful.* 

Every  one  .is  of  course  familiar  with  the  life  history  of  the 
butterfly,  how  it  lives  first  as  a  caterpillar  and  then  undergoes  a 
complete  transformation  into  a  winged  insect.  It  is  of  course 
evident  that  the  bodily  organs  of  the  caterpillar  are  not  at  all 
suited  to  the  needs  of  a  butterfly,  the  very  food  (of  those  species 
which  take  food)  being  entirely  different.  As  a  matter  of  fact, 
almost  every  portion  of  the  bodily  structure  of  the  butterfly  is 
dissolved,  as  it  were,  into  a  formless  pulp  at  the  beginning  of  the 
transformation,  and  the  organization  into  a  flying  insect  then 
grows  out  from  a  central  nucleus  very  much  as  a  chicken 
grows  in  the  food-stuff  of  an  egg.  So  it  is  in  the  growth  of  the 
civilized  man.     In  early  childhood,  if  the  individual  is  favored  by 

■5^  Let  not  the  reader  suppose  that  this  quality  of  detachment,  which  pervades  every 
branch  of  science,  and  most  of  all  the  science  of  physics,  confronts  the  specialist  only  ; 
it  is  a  universal  quality  and  it  stands  as  the  most  serious  obstacle  that  young  men  meet 
with  in  their  study  of  the  elementary  sciences. 


PREFACE.  vu 

Fortune,  he  exercises  and  develops,  more  or  less  extensively,  the 
primitive  instincts  of  the  race  in  a  free  outdoor  life,  and  the  ex- 
periences so  gained  are  so  much  mind-stuff  to  be  dissolved  and 
transformed  under  more  or  less  coercion  and  constraint  into  an 
effective  mind  of  the  twentieth  century  type. 

A  fourth  difficulty  is  that  the  possibility  of  rehabilitation  of 
mind-stuff  has  grown  up  as  a  human  faculty  almost  solely  on  the 
basis  of  language,  and  it  expresses  itself  almost  entirely  in  the 
formation  of  ideas,  whereas  a  great  deal  of  our  knowledge  of 
physics  is  correlated  in  mechanisms  and  can  scarcely  be  reduced 
to  verbal  forms  of  expression.  An  appreciative  friend  of  the 
senior  author,  upon  hearing  of  the  proposed  elementary  treatise 
on  physics,  by  Nichols  and  Franklin  in  1894,  expressed  his 
gratification  at  what  seemed  to  him  to  be  the  promise  of  a 
greatly  to  be  desired  development  of  the  whole  subject  of 
elementary  physics  out  of  the  idea  of  energy !  But  we  cannot 
get  away  from  machinery,*  and  the  mechanisms  of  manipulation 
and  measurement  are  essential  parts  of  the  structure  of  the 
physical  sciences  without  which,  the  whole  thing  degenerates  into 
an  ineffective  philosophy  of  weak  speculation. 

The  best  way  to  meet  this  quadruply  difficult  situation  is  to 
relate  the  teaching  of  the  physical  sciences  as  much  as  possible 
to  the  immediately  practical  things  of  life,  and,  repudiating  the 
repugnance  which  most  of  us  feel  for  "literary  style"  in  the 
handling  of  what  we  proudly  call  the  ''exact  sciences,"  go  in  for 
suggestiveness  as  the  only  way  to  avoid  the  total  inhibition  of 
the  httle  sense  that  is  born  with  our  students.  It  is  not,  how- 
ever, a  question  of  exactitude  versus  suggestiveness,  for  both  are 
necessary,  the  one  relating  chiefly  to  the  realm  of  ideas  and  the 
other  to  the  realms  of  objective  reality. f 

*See  Art.  134,  Chapter  XT. 

I  This  statement  is  purposely  left  somewhat  faulty  in  view  of  prevalent  ideas  as  to 
the  unity  of  the  mind.  Suggestiveness  is  the  one  form  of  appeal  to  the  marginal  con- 
tents of  the  mind,  and,  except  insofar  as  these  marginal  contents  are  elemental  in 
character,  suggestiveness  relates  no  more  distinctly  to  objective  reality  than  do  the 
precise  ideas  which  operate  almost  wholly  within  the  field  of  consciousness. 


viii  PREFACE. 

Such  a  method  is  certainly  calculated  to  '  limber  up  our  theories 
and  put  them  all  at  work,'  the  pragmatic  J  method  our  friends 
the  philosophers  call  it,  *  a  method  which  pretends  to  a  conquer- 
ing destiny '  ;  and  whatever  one  may  think  of  that  philosophy  of 
life  which  exhibits  itself  in  a  temperament  of  the  most  intensely 
practical  and  matter-of-fact  type,  it  is  certain  that  pragmatism  is 
the  only  philosophy  a  science  teacher  can  entertain  and  escape 
what  to  him  is  the  most  dangerous  form  of  idolatry,  science  for 
its  own  sake. 


We  wish  to  place  on  record  our  contempt  for  the  poundal  as 
an  actual  practical  unit  of  force,  and  also  for  the  *'slug"  (^2.2 
pounds)  as  an  actual  practical  unit  of  mass.  Engineers  will  per- 
haps always  measure  breaking  strengths  and  the  like  in  pounds- 
weight,  and  hungry  people  will  perhaps  always  buy  bread  and 
meat  by  the  pound.  But  most  certainly  we  do  need  the  word 
poundal  or  the  word  slug  so  as  to  enable  systematic  units  and 
practical  units  to  be  connected  in  intelligible  argument.  We  pre- 
fer the  word  poundal  for  this  purpose.  There  is  no  doubt  in  our 
minds  that  practical  units  should  be  accepted  in  their  entirety, 
pounds  of  cargo  and  pounds-weight  of  propelling  force,  and  we 
consider  that  to  attempt  to  "  simplify  "  the  equations  of  dynamics 
by  the  introduction  of  an  unfamiliar  and  unused  unit  of  force  or 
mass  is  ridiculous. 

In  reading  over  the  advanced  proofs  of  the  earlier  chapters  of 
this  book,  the  authors  become  painfully  aware  of  the  fact  that 
some  fifty  pages  of  introductory  matter  precede  what  is  ordinarily 
considered  to  be  the  proper  beginning  of  the  study  of  elementary 
mechanics ;  but  surely  no  one  can  question  the  necessity  of 
Chapter  II  on  Physical  Measurement,  and  if  anyone,  in  using  the 
book,  wishes  to  omit  Chapter  III  on  Physical  Arithmetic,  let  him 
do  so ;  we  cannot.  For  the  succeeding  chapters  we  offer  no 
apology,  indeed,  we  are  inclined  in  an  opposite  direction,  even  to 

J  From  the  Greek  word  7rpay/ia,  meaning  action,  from  which  our  words  *  practice  ' 
and  *  practical '  come. 


PREFACE. 


IX 


the  extent  of  emphasizing  the  importance  of  the  chapter  on  Wave 

Motion  and  Oscillatory  Motion. 

The  authors'  acknowledgments  are  due  to  Professor  J.  F.  Klein 

for  many  helpful  suggestions,  and  to  Mr.  J.  H.  Wily  for  the  care 

and  promptness  with  which  the  illustrations  have  been  prepared. 

W.  S.  Franklin, 

Barry  MacNutt. 
South  Bethlehem,  Pa. 
April  22,  1907. 


SYMBOLS. 

(This  list  includes  only  those  symbols  which  are  often  used.) 
a,  A,  linear  acceleration. 

a,  angular  acceleration. 

B,  bulk  modulus  of  a  substance. 

yS,  longitudinal  strain. 
d,  Dy  diameter,  or  distance,  or  density. 

E,  stretch  modulus  of  a  substance. 

Ff  force. 

g,  acceleration  of  gravity. 
ky  K,  a  proportionality  factor. 

K,  moment  of  inertia. 
/,  Z,  length. 
M,  m,  mass. 

/I,  coefficient  of  friction. 

71,  number  of  revolutions  per  second,  or  number  of  vibrations 
per  second.    Also  used  for  slide  modulus  of  a  substance. 

/,  hydrostatic  pressure. 

P,  power,  or  longitudinal  stress. 

<^,  angle  of  shear. 
r,  Ry  radius. 

5,  shearing  strain. 
ty  time. 

Ty  torque. 

T,  period  of  one  oscillation  of  a  vibrating  body. 
Vy  V,  linear  velocity,  or  volume. 

Wy  work  or  energy.     Also  used  for  weight. 
Xy  X,  an  abscissa,  or  a  horizontal  distance  from  a  fixed  point. 
y,  V,  an  ordinate,  or  a  vertical  distance  from  a  fixed  point. 

CO,  angular  velocity. 


TABLE   OF   CONTENTS. 

CHAPTER   I. 
Introduction i-  i/ 

CHAPTER   II. 
The  Measurement  of  Length,  Angle,  Mass  and  Time 18-32 

CHAPTER  III. 
Physical  Arithmetic 33-  53 

CHAPTER  IV. 
Simple  Statics 54-  64 

CHAPTER   V. 
Translatory  Motion 65-104 

CHAPTER  VI. 
Friction,  Work  and  Energy 105-127 

CHAPTER  VII. 
Rotatory  Motion 128-162 

CHAPTER  VIII. 
Elasticity  (Statics) 163-197 

CHAPTER  IX. 
Hydrostatics 198-217 

CHAPTER   X. 
Hydraulics 218-247 

CHAPTER  XI. 

Wave  Motion  and  Oscillatory  Motion 248-278 

Index •    ....  279-283 

xi 


THE  ELEMENTS  OF  MECHANICS, 


CHAPTER   I. 


INTRODUCTION. 


The  Laws  of  Motion.     Physical  Measurement.     The 
Study  of  Physics. 

At  the  beginning  of  this  study  of  mechanics  it  is  desirable  to 
establish  a  point  of  view  from  which  the  student  must  inevitably 
proceed  to  a  minute  consideration  of  the  bare  principles  of  the 
science,  and  in  attempting  to  accomplish  this  feat  of  insinuation 
the  senior  author  begs  leave  to  address  the  student  directly  as 
"my  young  friend,"  but  not,  indeed,  without  some  sense  of  the 
ludicrous  element  that  lurks  in  what  would  seem  to  be  an  evident 
presumption,  for  the  science  of  mechanics  is  not  supposed  to  be 
a  place  of  pleasant  paths  ! 

Perhaps  every  student  of  engineering  realizes  the  fact  that  his 
necessarily  specialized  course  of  study  is  apt  to  lead  him  to  a 
rather  narrow  view  of  human  life  as  a  whole.  Every  person  who 
studies  to  work,  and  few  indeed  are  those  who  should  not,  is  sure 
to  fall  short  in  this  respect.  It  is  important,  however,  that  every 
engineering  student  should  realize  two  things  in  connection  with 
his  study  of  the  physical  sciences.  The  first  is  that  the  study  of 
the  physical  sciences  is  exacting  beyond  all  compromise,  involving 
as  it  does  a  degree  of  coercion  and  constraint  which  it  is  beyond 
the  power  of  any  teacher  greatly  to  mitigate ;  and  the  second  is 


2  ELEMENTS   OF   MECHANICS. 

that  the  completest  science  stands  abashed  before  the  infinitely 
complicated  and  fluid  array  of  phenomena  of  the  material  world, 
except  only  in  the  assurance  which  its  method  gives. 

The  method  of  science ;  that,  my  young  friend,  is  where  con- 
straint and  exactions  lie,  and  what  I  would  say  to  you  concerning 
the  change  in  your  mode  of  thought  which  this  constraint  alone  can 
bring  about,  and  which  must  take  place  before  you  can  enter  into 
practical  life,  may  best  be  expressed  by  referring  to  the  life  his- 
tory of  a  tremafkable  little  animal,  the  axolotl,  a  kind  of  salaman- 
der cliat.ihes  a  tadpole-like  youth  and  never  changes  to  the 
adult  form  unless  a  stress  of  dry  weather  annihilates  his  watery 
world ;  but  he  lives  always,  and  reproduces  his  kind  as  a  tad- 
pole ;  and  a  very  odd  looking  tadpole  he  is  with  his  lungs 
hanging  from  the  sides  of  his  head  as  two  large  feathery  tassels, 
brownish  in  tint  from  the  red  blood  inside  and  the  mud  that 
settles  on  the  outside  of  the  tiny  tube -like  streamers.  When  the 
aquatic  home  of  the  axolotl  dries  up,  however,  he  quickly  de- 
velops a  pair  of  internal  lungs,  lops  off  his  tassels,  and  embarks 
on  a  new  mode  of  life  on  land.  If  you,  my  young  friend,  are  to 
develop  beyond  the  tadpole  stage,  you  must  meet,  with  quick 
and  responsive  inward  growth,  that  new  and  increasing  stress  of 
dryness,  as  many  men  are  wont  to  call  our  modern  age  of  science 
and  industry. 

Science,  as  young  people  study  it,  has  two  chief  aspects,  or,  in 
other  words,  it  may  be  roughly  divided  into  two  parts,  namely, 
the  study  of  those  things  which  come  upon  us^  as  it  were,  and  the 
study  of  those  things  which  we  deliberately  devise.  The  things 
that  come  upon  us  include  weather  phenomena  and  every  aspect 
and  phase  of  the  natural  world,  the  things  we  cannot  escape ; 
and  the  things  we  devise  relate  chiefly  to  the  serious  work  of 
the  world,  the  things  we  laboriously  build,  and  the  things  we 
deliberately  and  patiently  seek. 

You  are  of  course  familiar  with  the  accepted  division  of  science 
into  biology,  the  science  of  plants  and  animals,  and  physics,  the 
science  of  machines  (what  is  a  telescope,  a  test  tube,  or  a  still, 


INTRODUCTION.  3 

but  a  machine  ?).  To  a  certain  extent  this  corresponds  to  the 
division  of  science  on  the  basis  of  what  comes  upon  us  and  what 
we  devise ;  but,  strictly  speaking,  this  division  of  science  into 
biology  and  physics  has  but  little  to  do  with  the  distinction  be- 
tween the  things  that  come  upon  us  and  the  things  we  devise ; 
and  yet  it  is  necessary  for  our  present  insinuating  purpose  to  di- 
vide science  on  the  basis  of  something  which  stands  out  sharply 
and  distinctly  in  the  experiences  of  everyday  life,  so  as  to  leave 
in  each  fraction  as  much  as  possible  of  the  vital  qualities  that 
give  pleasure  and  exact  pains.  We  will  therefore,  for  the  present, 
divide  science  on  the  basis  of  work  and  —  play,  if  I  may  be  allowed 
so  to  mark  the  contrast  between  the  pranks  of  the  gods  and  the 
labors  of  men. 

The  Laws  of  Motion. 

The  most  prominent  aspect  of  all  phenomena  is  motion.  In 
that  realm  of  nature  which  is  not  of  man's  devising,  motion  is 
universal ;  ripple  of  brook,  and  gentle  flow  of  river,  ocean's 
swell,  and  slow  recurrent  tide,  and  over  all  an  incessant  wander- 
ing waft  of  wind,  infinitely  varied.  In  the  other  realm  of  nature, 
the  realm  of  things  devised,  motion  is  no  less  prominent.  Every 
purpose  of  our  practical  Hfe  is  accompHshed  by  movements  of 
the  body  and  by  directed  movements  of  tools  and  mechanisms, 
such  as  the  swing  of  the  scythe  and  flail,  and  those  studied 
movements  of  planer  and  lathe  from  which  are  evolved  strong 
arm  of  steam  shovel  and  deft  fingers  of  the  loom,  the  writing 
hand  of  the  telegraph  and  tongue  of  the  telephone,  and  tireless 
travelers  that  have  no  feet  nor  back  to  burden  with  heavy  load. 

The  laws  of  motion  !  You,  my  young  friend,  must  have  in 
some  measure  my  own  youthful  view,  which,  to  tell  the  truth, 
I  have  never  wholly  lost,  that  there  is  something  absurd  in  the 
idea  of  reducing  the  more  complicated  phenomena  of  nature  to 
any  orderly  system  of  mechanical  law.  For,  to  speak  of  motion 
is  no  doubt  to  call  to  your  mind  first  of  all  the  phenomena  that 
are   associated    with    the    excessively    complicated,    incessantly 


4  ELEMENTS   OF   MECHANICS. 

changing,  turbulent  and  tumbling  motion  of  wind  and  water. 
These  phenomena  have  always  had  the  most  insistent  appeal  to 
us,  they  have  confronted  us  everywhere  and  always,  and  life  is 
an  unending  contest  with  their  fortuitous  diversity,  which  rises 
only  too  often  to  irresistible  sweeps  of  destruction  in  fire  and 
flood,  and  in  calamitous  crash  of  collision  and  collapse  where  all 
things  commingle  in  one  dread  fluid  confusion. 

The  laws  of  motion !  When  I  was  a  boy  I  read  many  scien- 
tific books  with  great  interest,  but  I  never  could  quite  reconcile 
myself  to  a  certain  arrogant  suggestion  of  completeness  and  uni- 
versaHty.  The  laws  of  motion  !  Truly  the  science  of  mechanics 
is  circumscribed  in  utter  repudiation  of  such  universal  phrase  ; 
and  yet,  curiously  enough,  the  ideas  which  constitute  the  laws  of 
motion  have  an  almost  unlimited  extent  of  legitimate  range,  and 
these  ideas  must  be  possessed  with  perfect  precision  if  one  is  to 
acquire  any  solid  knowledge  whatever  of  the  phenomena  of  mo- 
tion, especially  in  the  realm  of  devices  which  so  serviceably  ac- 
complish the  purposes  of  the  human  will,  for  in  this  realm,  at 
least,  everything  is  really  and  adequately  correlated  in  a  closely 
woven  fabric  of  reason. 

The  necessity  of  precise  ideas.  Herein  lies  the  impossibility  of 
compromise  and  the  necessity  of  coercion  and  constraint ;  one 
must  think  so  and  so,  there  is  no  other  way.  And  the  realm  of 
precise  ideas,  that  is  the  region  I  intended  to  symbolize  by  the 
land  whereon  the  little  tasseled  tadpole,  the  axolotl,  is  forced  to 
live  by  unwelcome  stress  of  weather.  I  remember  as  a  boy  a 
sharp  contest  in  my  own  mind  between  an  extremely  vivid  sense 
of  things  physical  and  the  constraining  function  of  precise  ideas. 
This  contest  is  perennial,  and  it  is  by  no  means  a  onesided  con- 
test between  mere  crudity  and  refinement,  for  refinement  ignores 
many  things.  Indeed,  precise  ideas  not  only  help  to  form  our 
sense  of  the  world  in  which  we  live,  but  they  inhibit  sense  as 
well,  and  their  rigid  and  unchallenged  rule  would  indeed  be  a 
stress  of  dryness. 

The  laws  of  motion.     I  return  again  and  yet  again  to  my  sub- 


INTRODUCTION.  5 

ject,  for  indeed  I  am  not  to  be  deterred  therefrom  by  any  con- 
cession of  inadequacy,  no,  nor  by  any  degree  of  realization  of  the 
vividness  of  your  youthful  sense  of  those  things  which,  to  suit 
my  narrow  purpose,  must  be  stripped  completely  bare.  It  is  un- 
fortunate, however,  for  my  purpose  that  the  prevailing  type  of 
motion,  the  flowing  of  water  and  the  blowing  of  the  wind,  is  be- 
wilderingly  useless  as  a  basis  for  the  establishment  of  the  simple 
and  precise  ideas  which  are  called  the  laws  of  motion,  and  which 
are  the  most  important  of  the  fundamental  principles  of  physics. 
These  ideas  have,  in  fact,  grown  out  of  the  study  of  the  simple 
phenomena  which  are  associated  with  the  motion  of  bodies  in 
bulk,  without  perceptible  change  of  form,  the  motion  of  rigid 
bodies,  so  called  ;  and  these  ideas  are  limited  in  their  strict  appli- 
cation to  these  simple  phenomena. 

Before  narrowing  down  the  scope  of  my  discussion,  however, 
let  me  illustrate  a  very  general  application  of  the  simplest  idea 
of  motion,  the  idea  of  velocity.  You  have,  no  doubt,  an  idea  of 
what  is  meant  by  the  velocity  of  the  wind ;  and  a  sailor,  having 
what  he  calls  a  ten-knot  wind,  knows  that  he  can  manage  his 
boat  with  a  certain  spread  of  canvas  and  that  he  can  accomplish 
a  certain  portion  of  his  voyage  in  a  given  time  ;  but  an  experi- 
enced sailor,  although  he  speaks  glibly  of  a  ten-knot  wind,  belies 
his  speech  by  taking  wise  precaution  against  every  conceivable 
emergency.  He  knows  that  a  ten-knot  wind  is  by  no  means  a 
sure  or  a  simple  thing,  with  its  incessant  blasts  and  whirls  ;  and  a 
sensitive  anemometer,  having  more  regard  for  minutiae  than  any 
sailor,  usually  registers  in  every  wind  a  number  of  almost  com- 
plete but  excessively  irregular  stops  and  starts  every  minute  and 
variations  of  direction  that  sweep  round  half  the  horizon  ! 

We  must,  as  you  see,  direct  our  attention  to  something  simpler 
than  the  wind.  Let  us  therefore  consider  the  drawing  of  a  wagon 
or  the  propulsion  of  a  boat.  It  is  a  familiar  experience  that 
effort  is  required  to  start  a  body  moving,  and  that  continued 
effort  is  required  to  maintain  the  motion.  Certain  very  simple 
facts  as  to  the  nature  of  this  effort,  as  to  the  amount  of  effort  re- 


6  ELEMENTS   OF   MECHANICS. 

quired  to  produce  motion,  and  as  to  the  conditions  which  deter- 
mine the  amount  of  effort  required  to  keep  a  body  in  motion 
were  discovered  by  Sir  Isaac  Newton  and  these  facts  are  called 
the  laws  of  motion. 

The  effort  required  to  start  a  body  or  to  keep  it  moving  is 
called  force.  Thus  if  I  start  a  box  sliding  along  a  table  I  am 
said  to  exert  a  force  on  the  box.  I  might  accomplish  the  same 
effect  by  interposing  a  stick  between  my  hand  and  the  box,  in 
which  case  I  would  exert  a  force  on  the  stick  and  the  stick  in  its 
turn  would  exert  a  force  on  the  box.  We  thus  arrive  at  the 
notion  of  force  action  between  inanimate  bodies,  between  the  stick 
and  the  box  in  this  case,  and  Newton  pointed  out  that  the  force 
action  between  two  bodies  A  and  B  always  consists  of  two  equal 
and  opposite  forces.  That  is  to  say,  if  body  A  exerts  a  force  on 
B,  then  B  exerts  an  equal  and  opposite  force  on  A,  or,  to  use 
Newton's  words,  action  is  equal  to  reaction  and  in  a  con- 
trary DIRECTION. 

I  might  have  led  up  to  a  statement  of  this  fact  by  considering 
the  force  with  which  I  push  on  the  box  and  the  equal  and  oppo- 
site force  with  which  the  box  pushes  back  on  me,  but  if  I  do  not 
wish  to  introduce  the  stick  as  an  intermediary,  it  is  much  better 
to  speak  of  the  force  with  which  my  hand  pushes  on  the  box  and 
the  equal  and  opposite  force  with  which  the  box  pushes  back  on 
my  hand,  because  in  discussing  physical  things  it  is  of  the 
utmost  importance  to  eliminate  the  personal  element.  I  do  not 
think  I  shall  find  a  better  opportunity  to  explain  to  you  further 
what  I  mean  by  the  reference  I  have  made  to  the  curious  life 
history  of  the  little  Mexican  salamander,  the  axolotl.  It  is  that 
our  modern  industrial  life,  in  bringing  men  face  to  face  with  an 
entirely  unprecedented  array  of  intricate  mechanical  and  physical 
problems,  demands  of  every  one  a  great  and  increasing  amount 
of  dry,  impersonal  thinking,  and  that  the  precise  and  rigorous 
modes  of  thought  of  the  modern  physical  sciences  are  being 
forced  upon  widening  circles  of  men  with  a  relentless  insistence 
which  must  soon  give  them  complete  and  universal  dominion  in 


INTRODUCTION.  7 

those  realms  of  thought  which  have  to  do  with  the  harder  and 
more  exacting  aspects  of  man's  relation  to  physical  things. 

When  we  examine  into  the  conditions  under  which  a  body- 
starts  to  move  and  the  conditions  under  which  a  body  once 
started  is  kept  in  motion,  we  shall  come  across  a  very  remarkable 
fact,  if  we  are  careful  to  consider  every  force  which  acts  upon  the 
body,  and  this  remarkable  fact  is  that  the  forces  which  act  upon  a 
body  which  remains  at  rest  are  related  to  each  other  in  precisely 
the  same  way  as  the  forces  which  act  upon  a  body  which  con- 
tinues to  move  steadily  along  a  straight  path.  Therefore,  it  is  . 
convenient  to  consider,  first  the  relation  between  the  forces  which 
act  upon  a  body  at  rest,  or  upon  a  body  in  uniform  motion,  and 
then  to  consider  the  relation  between  the  forces  which  act  upon 
a  body  which  is  starting  or  stopping  or  changing  the  direction 
of  its  motion. 

Suppose  you  were  to  hold  a  box  in  mid-air.  To  do  so  it 
would  of  course  be  necessary  for  you  to  push  up  on  the  box  so 
as  to  balance  the  downward  pull  of  the  earth,  the  weight  of  the 
box,  as  it  is  called.  Then  if  I  were  to  take  hold  of  the  box  and 
pull  upon  it  in  any  direction,  you  would  have  to  exert  an  equal 
pull  on  the  box  in  the  opposite  direction  to  keep  it  stationary. 
That  is  to  say,  the  forces  which  act  upon  a  stationary  body  are 
always  balanced. 

Every  one,  perhaps,  realizes  that  what  I  have  said  about  the 
balanced  relation  of  the  forces  which  act  upon  a  stationary  box, 
is  equally  true  of  the  forces  which  act  on  a  box  similarly  held  in 
a  steadily  moving  railway  car  or  boat.  Therefore,  the  forces 
which  act  upon  a  body  which  moves  steadily  along  a  straight 
path  are  balanced. 

This  is  evidently  true,  as  I  have  pointed  out,  when  the  mov- 
ing body  is  surrounded  on  all  sides  by  things  which  are  moving 
along  with  it,  as  a  car  or  a  boat ;  but  how  about  a  body  which 
moves  steadily  in  a  straight  path  but  which  is  surrounded  by 
bodies  which  do  not  move  along  with  it  ?  You  know  that  some 
active  agent  such  as  a  horse  or  a  steam  engine  must  pull  steadily 


8  ELEMENTS   OF   MECHANICS. 

upon  such  a  body  to  keep  it  in  motion,  and  you  know  that  if  left 
to  itself  such  a  moving  body  quickly  comes  to  rest.  No  doubt 
you  have  reached  this  further  inference  from  your  experience  that 
this  tendency  of  moving  bodies  to  come  to  rest  is  due  to  the  drag- 
ging forces,  or  friction,  with  which  surrounding  bodies  act  upon 
a  body  in  motion.  Thus  a  moving  boat  is  brought  to  rest  by  the 
drag  of  the  water  when  the  propelling  force  ceases  to  act ;  a  train 
of  cars  is  brought  to  rest  because  of  the  drag  due  to  friction 
when  the  pull  of  the  locomotive  ceases  ;  a  box  which  is  moved 
across  a  table  quickly  comes  to  rest  when  left  to  itself,  because 
of  the  drag  due  to  friction  between  the  box  and  the  table. 

We  must,  therefore,  always  consider  two  distinct  forces  when 
we  are  concerned  with  a  body  which  is  kept  in  motion,  namely, 
the  propelling  force  due  to  some  active  agent  such  as  a  horse  or 
an  engine,  and  the  dragging  force  due  to  surrounding  bodies. 
Newton  pointed  out  that  when  a  body  is  moving  steadily  along 
a  straight  path,  the  propelling  force  is  always  equal  and  opposite 
to  the  dragging  force.     Therefore,  the  forces  which  act  upon 

A  BODY  WHICH  IS  STATIONARY,  OR  WHICH  IS  MOVING  UNIFORMLY 
ALONG  A  STRAIGHT  PATH,  ARE  BALANCED  FORCES. 

I  imagine  that  you  will  hesitate  to  accept  as  a  fact  the  complete 
and  exact  balance  of  propelling  and  dragging  forces  on  a  body 
which  is  moving  steadily  along  a  straight  path  in  the  open.  All 
I  can  say  is  that  direct  experiment  shows  it  to  be  true,  and  that 
the  most  elaborate  calculations  and  inferences  based  upon  this 
notion  of  the  complete  balance  of  propelling  and  dragging  forces 
on  a  body  in  uniform  motion  are  verified  by  experiment.  You 
may  ask,  why  should  a  canal  boat,  for  example,  continue  to 
move  if  the  pull  of  the  horse  does  not  exceed  the  drag  of  the 
water ;  but  why  should  it  stop  if  the  drag  does  not  exceed  the 
pull  ?  You  understand  that  we  are  not  considering  the  starting 
of  the  boat.  The  fact  is  that  the  conscious  effort  which  one  must 
exert  even  to  drive  a  horse,  the  cost  of  the  horse,  and  the  ex- 
pense of  his  keep,  are  what  most  people  think  of,  however  hard 
one  tries  to  direct  their  attention  solely  to  the  state  of  tension  in 


INTRODUCTION.  9 

the  rope  that  hitches  the  horse  to  the  boat  after  the  boat  is  in  full 
motion  ;  and  most  people  come  upon  the  idea  that  if  the  function 
of  the  horse  is  simply  to  balance  the  drag  of  the  water  so  as  to 
keep  the  boat  from  stopping,  then  why  should  there  not  be  some 
way  to  avoid  the  cost  of  so  insignificant  an  operation  ?  There  is, 
indeed,  an  extremely  important  matter  involved  here  which  we 
will  consider  when  we  come  to  the  discussion  of  work  and  energy ; 
but  it  has  no  bearing  on  the  matter  of  the  balance  of  propulsion 
and  drag  on  a  body  which  moves  steadily  along  a  straight  path. 
Let  us  now  consider  the  relation  between  the  forces  which  act 
upon  a  body  which  is  changing  its  speed,  upon  a  body  which  is 
being  started  or  stopped,  for  example.  I  suppose  that  you  have 
noticed  how  a  horse  strains  at  his  rope  when  starting  a  canal  boat, 
especially  if  the  boat  is  heavily  loaded,  and  how  the  boat  con- 
tinues to  move  for  a  long  time  after  the  horse  ceases  to  pull.  In 
the  first  case,  the  pull  of  the  horse  greatly  exceeds  the  drag  of  the 
water,  and  the  speed  of  the  boat  increases ;  and  in  the  second 
case,  the  drag  of  the  water  of  course  exceeds  the  pull  of  the  horse, 
for  the  horse  is  not  pulling  at  all,  and  the  speed  of  the  boat  de- 
creases. When  the  speed  of  a  body  is  changing,  the  forces  which 
act  on  the  body  are  unbalanced,  and  we  may  conclude  that  the 
effect  of  an  unbalanced  force  acting  on  a  body  is  to  change  the 
velocity  of  the  body ;  and  it  is  evident  that  the  longer  the  unbal- 
anced force  continues  to  act  the  greater  the  change  of  velocity. 
Thus  if  the  horse  ceases  to  pull  on  a  canal  boat  for  one  Seconal  the 
velocity  of  the  boat  will  be  but  slightly  reduced  by  the  unbalanced 
drag  of  the  water,  whereas  if  the  horse  ceases  to  pull  for  two  seconds 
the  decrease  of  velocity  will  be  much  greater.  In  fact  the  change 
of  velocity  due  to  a  given  unbalanced  force  is  proportional  to  the 
time  that  the  force  continues  to  act.  This  is  exemplified  by  a  body 
falling  under  the  action  of  the  unbalanced  pull  of  the  earth  ;  after 
one  second  it  will  have  gained  a  certain  amount  of  velocity  (about 
32  feet  per  second),  after  two  seconds  it  will  have  made  a  total 
gain  of  twice  as  much  velocity  (about  64  feet  per  second),  and  so 
on.      Furthermore,  since  the  velocity  produced  by  an  unbalanced 


lO  ELEMENTS   OF   MECHANICS. 

force  is  proportional  to  the  time  that  the  force  continues  to  act, 
it  is  evident  that  the  effect  of  the  force  should  be  specified  as  so 
much  velocity  produced  per  second,  exactly  as  in  the  case  of 
earning  money,  the  amount  one  earns  is  proportional  to  the  length 
of  time  that  one  continues  to  work,  and  we  always  specify  one's 
earning  capacity  as  so  much  money  earned  per  day. 

Everyone  knows  what  it  means  to  give  an  easy  pull  or  a  hard 
pull  on  a  body.  That  is  to  say,  we  all  have  the  ideas  of  greater 
and  less  as  applied  to  forces.  Everybody  knows  also  that  if  a 
horse  pulls  hard  at  a  canal  boat,  the  boat  will  get  under  way  more 
quickly  than  if  the  pull  is  easy,  that  is,  the  boat  will  gain  more 
velocity  per  unit  of  time  under  the  action  of  a  hard  pull  than 
under  the  action  of  an  easy  pull.  Therefore,  any  precise  state- 
ment of  the  effect  of  an  unbalanced  force  on  a  given  body  must 
correlate  the  precise  value  of  the  force  and  the  exact  amount  of 
velocity  produced  per  unit  of  time  by  the  force.  This  seems  a 
very  difficult  thing,  but  its  apparent  difficulty  is  very  largely  due 
to  the  fact  that  as  yet  we  have  not  agreed  as  to  what  we  are  to 
understand  by  the  statement  that  one  force  is  precisely  three,  or 
four,  or  any  number  of  times  as  great  as  another.  Suppose,  there- 
fore, that  we  agree  to  call  one  force  twice  as  large  as  another 
when  it  will  produce  in  a  given  body  twice  as  much  velocity  in  a 
given  time  (remembering  of  course  that  we  are  now  talking  about 
unbalanced  forces,  or  that  we  are  assuming  for  the  sake  of  sim- 
plicity of  statement,  that  no  dragging  forces  exist).  As  a  result 
of  this  definition  we  may  state  that  the  amount  of  velocity  pro- 
duced per  second  in  a  given  body  by  an  unbalanced  force  is  pro- 
portional to  the  force. 

Of  course  w^e  know  no  more  about  the  matter  in  hand  than  we 
did  before  we  adopted  the  definition,  but  we  do  have  a  good  illus- 
tration of  how  important  a  part  is  played  in  the  study  of  science, 
by  what  we  may  call  making  up  one's  mind,  in  the  sense  of  put- 
ting one's  mind  in  order.  This  kind  of  thing  is  very  prominent 
in  the  study  of  elementary  physics,  and  by  that  rather  indefinite 
reference,  in  my  story  of  the  little  tasseled  tadpole,  to  an  inward 


INTRODUCTION.  II 

growth  so  needful  to  you  before  you  can  hope  for  any  measure 
of  success  in  our  modern  world  of  scientific  industry,  I  meant  to 
refer  to  this  thing,  the  *'  making-up  "  of  one's  mind.  Nothing  is 
so  essential  in  the  acquirement  of  exact  and  solid  knowledge  as 
the  possession  of  precise  ideas,  not  indeed  that  a  perfect  precision 
is  necessary  as  a  means  for  retaining  knowledge,  but  that  nothing 
else  so  effectually  opens  the  mind  for  the  perception  even  of  the 
simplest  evidences  of  a  subject.^  We  may  illustrate  these  things 
further  by  following  up  our  discussion  of  the  laws  of  motion. 

We  have  now  settled  the  question  as  to  the  effect  of  different 
unbalanced  forces  on  a  given  body  on  the  basis  of  a  very  general 
experience,  and  by  an  agreement  as  to  the  precise  meaning  to  be 
attached  to  the  statement  that  one  force  is  so  many  times  as 
great  as  another ;  but  how  about  the  effect  of  a  given  force  upon 
different  bodies,  and  how  may  we  identify  the  force  so  as  to  be 
sure  that  it  is  the  same  ?  As  to  the  identification,  a  given  force 
may  be  made  to  act  on  any  body  by  causing  a  given  body  to 
exert  the  force,  and  by  considering  whether  the  reaction  produces 
the  same  effect  on  the  given  body  in  each  case.  Thus  a  spring 
dynamometer  may  be  used  to  exert  a  given  force  on  any  body, 
the  reaction  on  the  spring  dynamometer  causing  a  given  stretch 
of  the  spring.  As  to  the  effect  of  a  given  unbalanced  force  in 
producing  velocity  of  different  bodies,  three  things  have  to  be 
settled  by  experiment. 

{a)  In  the  first  place  let  us  suppose  that  a  certain  force  A  is 
twice  as  large  as  a  certain  other  force  B,  according  to  our  agree- 
ment, because  the  force  A  produces  twice  as  much  velocity  every 
second  as  force  B  when  the  one  and  then  the  other  of  these  forces 
is  caused  to  act  upon  a  given  body,  a  piece  of  lead,  for  example. 
Then,  will  the  force  A  produce  twice  as  much  velocity  every  second 
as  the  force  B  whatever  the  nature  and  size  of  the  body,  whether 
it  be  wood,  or  ice,  or  sugar  ?     Experiment  shows  that  it  will. 

*  Opens  the  mind,  that  is,  for  those  things  which  are  conformable  to  or  consistent 
with  the  ideas.  The  history  of  science  presents  many  cases  in  which  accepted  ideas 
have  closed  the  mind  to  contrary  evidences  for  many  generations.  Let  young  men 
beware  ! 


12-  ELEMENTS   OF   MECHANICS. 

{b)  In  the  second  place,  suppose  that  we  have  such  amounts 
of  lead,  of  iron,  of  wood,  etc.,  that  a  certain  given  force  produces 
the  same  amount  of  velocity  per  second  when  it  is  made  to  act, 
as  an  unbalanced  force,  upon  one  or  another  of  these  various 
bodies.  Then  what  is  the  relation  between  the  amounts  of  these 
various  substances  ?  Experiment  shows  that  they  all  have  the 
same  mass  in  grams,  or  pounds,  as  determined  by  a  balance.  That 
is,  a  given  force  produces  the  same  amount  of  velocity  per  second 
in  a  given  number  of  grams  of  any  kind  of  substance.  Thus  the 
earth  pulls  with  a  certain  definite  force  (in  a  given  locality)  upon 
M  grams  of  any  substance  and,  aside  from  the  dragging  forces 
due  to  air  friction,  all  kinds  of  bodies  gain  the  same  amount 
of  velocity  per  second  when  they  fall  under  action  of  the  unbal- 
anced pull  of  the  earth. 

(c)  In  the  third  place,  what  is  the  relation  between  the  velocity 
per  second  produced  by  a  given  force  and  the  mass  in  grams 
of  the  body  upon  which  it  acts  ?  Experiment  shows  that  the 
velocity  per  second  produced  by  a  given  force  is  inversely  pro- 
portional to  the  mass  of  the  body  upon  which  the  force  acts. 

The  effect  of  an  unbalanced  force  in  producing  velocity  may 
therefore  be  summed  up  as  follows :  The  velocity  per  second 

PRODUCED    BY    AN    UNBALANCED    FORCE    IS    PROPORTIONAL   TO    THE 
FORCE  AND    INVERSELY   PROPORTIONAL  TO    THE    MASS  OF  THE  BODY 

UPON  WHICH  THE  FORCE  ACTS.      Furthermore,  the  velocity  pro- 
duced BY  AN  UNBALANCED  FORCE  IS  ALWAYS  IN  THE  DIRECTION  OF 

the  force. 

Physical  Measurement. 

Among  primitive  races  all  things  subject  to  exchange  or  bar- 
ter are  estimated  by  simple  counting.  Thus  a  Tartar  herds- 
man estimates  his  wealth  by  counting  his  cattle.  With  the 
growth  of  civilization,  however,  there  has  been  a  great  increase 
in  the  variety  of  useful  and  exchangeable  commodities,  and  many 
of  these  commodities,  molasses,  for  example,  cannot  be  estimated 
by  simple  counting.  The  result  has  been  that  the  simple  opera- 
tion of  counting,  which,  of  course,  can  be  applied  only  to  groups 


INTRODUCTION.  1 3 

of  separate  and  distinct  things,  has  developed  into  the  operation 
called  measurement,  in  which  a  continuous  whole  is  estimated 
numerically  by  dividing  it  into  equal,  unit  parts,  and  counting 
these  parts.  Thus  oil  or  wine  is  counted  out  by  means  of  a  gal- 
lon measure,  and  cloth  is  counted  out  by  means  of  a  yard-stick. 

In  many  kinds  of  measurement,  the  two  distinct  operations, 
(a)  dividing  into  equal  unit  parts  and  (U)  counting  are  obscured  by 
the  use  of  more  or  less  elaborate  measuring  devices,  but  every 
measurement  of  whatever  kind  does,  in  fact,  consist  of  these  two 
fundamental  operations.  Thus  in  measuring  a  length  by  means 
of  a  scale  of  inches,  the  operation  of  dividing  into  unit  parts  has 
been  performed  once  for  all  by  the  maker  of  the  scale,  and  in 
this  case  the  operation  of  counting  is,  in  large  part,  "ready-made  " 
by  the  numbers  stamped  on  the  scale.  In  the  weighing  of  a  con- 
signment of  coal  or  iron,  the  operation  of  dividing  into  unit  parts 
has  been  performed  once  for  all  by  the  maker  of  the  set  of 
weights  and  of  the  divided  balance  beam,  and  the  operation  of 
counting  is,  in  large  part,  "■  ready-made  "  by  the  numbers  stamped 
upon  the  weights  and  upon  the  beam. 

The  long  experience  of  the  race  in  estimating  by  the  simple 
counting  of  separate  things  has  given  rise  to  a  sense  of  sharp  dis- 
tinction between  any  two  numbers,  thus  1000  horses  is  clearly 
not  the  same  thing  as  999  horses  ;  but  this  sharp  distinction  be- 
tween approximately  equal  numbers  is  devoid  of  physical  signifi- 
cance in  the  case  of  numbers  derived  by  measurement,  because  of 
the  approximate  character  of  the  operation  of  dividing  a  whole  ijtto 
unit  parts.  A  person  might  buy  a  herd  of  horses  supposing  the 
number  to  be  1000,  whereas  a  correct  count  would  show  999; 
and  although  the  purchaser  might  reasonably  say,  "■  Oh,  let  it  go 
it  makes  no  difference,"  still  the  fact  would  remain  that  999 
horses  is  not  1 000  horses  ;  but  suppose  a  man  were  to  buy 
1000  yards  of  cloth,  he  might  remeasure  the  cloth  and  count 
999  yards,  but  in  remeasuring  the  day  may  have  been  damp,  o** 
he  may  not  have  stretched  the  cloth  in  the  same  way  as  the  man- 
ufacturer, or  he  may  have  taken  more  or  less  pains  in  fitting  the 


14  ELEMENTS   OF   MECHANICS. 

yard-stick  to  the  successive  portions  of  the  cloth,  or  his  yard- 
stick may  have  been  in  error.  The  fact  is  that  it  is  impossible  to 
show  that  looo  yards  of  cloth  is  not  999  yards  of  cloth,  except 
by  reasoning  that  1000  pieces  of  silver  is  not  999  pieces  of  silver. 
The  difficulty  is  that  a  yard  of  cloth  is  not  a  separate  thing 
whereas  a  piece  of  silver  is. 

Nothing  is  more  amusingly  indicative  of  a  disregard  of  physi- 
cal facts  than  to  see  a  long  array  of  digits  carried  laboriously 
through  an  arithmetical  calculation  which  is  based  upon  numeri- 
cal data  obtained  by  physical  measurement.  For  example,  a 
man  weighs  a  body  in  air  and  then  suspends  it  by  a  string  under 
water  and  weighs  it  again,  obtaining  105.26  grams  and  74.63 
grams  respectively  ;  and  from  these  data  he  calculates  the  specific 
gravity  to  be  3.436500489  -f-.  This  would  be  sufficiently  amus- 
ing if  it  were  certain  that  the  two  numbers  105.26  and  74.63 
were  free,  as  far  as  they  go,  from  every  influence  like  those  de- 
scribed above  as  modifying  the  measured  length  of  a  piece 
of  cloth ;  but  if  this  is  not  certain,  then  the  thing  is  indeed 
ridiculous. 

The  operation  of  dividing  a  length  or  an  angle  into  equal  unit 
parts  for  the  purpose  of  measurement,  is  an  operation  oi fitting  a 
standard  to  each  part,  an  operation  of  congruence ;  and  the 
actual  measurement  of  any  physical  whole  —  let  us  not  speak  of 
it  as  a  quantity  until  we  have  attached  a  number  to  it  —  depends 
upon  one  or  another  variety  of  congruence  as  a  basis  for  the 
assumption  of  equality  of  the  parts  which  are  to  be  counted. 
Thus  a  pendulum  may  be  assumed  to  mark  off  equal  intervals 
of  time  because  each  movement  of  the  pendulum  is  like  the  one 
that  follows  ;  and  the  equal  arm  balance  is  a  device  for  indicating 
a  certain  kind  of  congruence  between  the  body  which  is  being 
weighed  and  the  combination  of  weights  which  balances  it. 

The  fundamental  meaning  of  a  physical  quantity  originates  in 
and  is  defined  by  the  actual  operation  of  measuring  that  quantity. 
Thus  it  is  sheer  nonsense  to  define  the  mass  of  a  body  as  **  the 
amount  of  material  the  body  contains."     The  mass  of  a  body, 


INTRODUCTION.  1 5 

as  a  quantity,  is  defined  by  the  operation  of  weighing  by  a  bal- 
ance ;  and,  since  the  result  of  this  operation  is  always  the  same, 
within  the  limits  of  error,  for  a  given  amount  of  any  substance, 
it  is  permissible  to  use  this  result  as  a  measure  of  the  amount  of 
the  substance.  Nearly  every  physical  definition^  rightly  understood^ 
is  an  actual  physical  operation. 

The  Science  of  Physics. 

"  We  advise  all  men  "  says  Bacon  "  to  think  of  the  true  ends 
of  knowledge,  and  that  they  endeavor  not  after  it  for  curiosity, 
contention,  or  the  sake  of  despising  others,  nor  yet  for  reputation 
or  power  or  any  other  such  inferior  consideration,  but  solely  for 
the  occasions  and  uses  of  life."  It  is,  indeed,  impossible  to 
imagine  any  other  basis  upon  which  the  study  of  physics  can  be 
justified  than  for  the  occasions  and  uses  of  life.  At  any  rate, 
more  than  nine-tenths  of  the  subject  matter  of  physics  now 
relates  to  the  conditions  which  have  been  elaborated  through  the 
devices  of  industry,  and  the  study  of  physics  has  to  do  almost 
wholly  with  devised  phenomena,  as  exemplified  in  our  mills  and 
factories,  in  our  machinery  of  transportation,  in  optical  and  musical 
instruments,  in  the  means  for  the  supply  of  power,  heat,  light, 
and  water  for  general  and  domestic  use,  and  so  on. 

From  this  extremely  practical  point  of  view  it  may  seem  that 
there  can  be  nothing  very  exacting  in  the  study  of  the  physical 
sciences  ;  but  what  is  physics  ?  That  is  the  question.  One  defini- 
tion at  least  we  must  repudiate  ;  it  is  not  "  The  science  of  masses, 
molecules,  and  the  ether."  Bodies  have  mass  and  railways  have 
length,  and  to  speak  of  physics  as  the  "  science  of  masses  "  is 
as  silly  as  to  define  railroading  as  the  ''practice  of  lengths,"  and 
nothing  as  reasonable  as  this  can  be  said  in  favor  of  the  concep- 
tion of  physics  as  the  science  of  molecules  and  the  ether ;  it  is 
the  sickliest  possible  notion  of  physics  even  if  a  student  really 
gets  it,  whereas  the  healthiest  notion,  even  if  a  student  does  not 
wholly  grasp  it,  is  that  physics  is  the  science  of  the  ways  of  tak- 
ing hold  of  things  and  pushing  them  ! 


l6     .  ELEMENTS   OF   MECHANICS. 

Bacon  long  ago  listed  in  his  quaint  way  the  things  which  seemed 
to  him  most  needful  for  the  advancement  of  learning.  Among 
other  things  he  mentioned  *'  A  New  Engine  or  a  Help  to  the 
mind  corresponding  to  Tools  for  the  hand,"  and  the  most  re- 
markable aspect  of  physical  science  is  that  aspect  in  which  it  con- 
stitutes a  realization  of  this  New  Engine  of  Bacon.  We  con- 
tinually force  upon  the  extremely  meager  data  obtained  directly 
through  our  senses,  an  interpretation  which,  in  its  complexity  and 
penetration,  would  seem  to  be  entirely  incommensurate  with  the 
data  themselves,  and  the  possibility  of  this  forced  interpretation 
depends  upon  the  use  of  two  complexes  :  (a)  A  logical  structure, 
that  is  to  say,  a  body  of  mathematical  and  conceptual  theory  which 
is  brought  to  bear  upon  the  immediate  materials  of  sense,  and  (b) 
a  mechanical  structure,  that  is  to  say,  either  (i)  a  carefully  planned 
arrangement  of  apparatus,  such  as  is  always  necessary  in  making 
physical  measurements,  or  (2)  a  carefully  planned  order  of  opera- 
tions, such  as  the  successive  operations  of  solution,  reaction,  pre- 
cipitation, filtration,  and  weighing  in  chemistry. 

These  two  complexes  do  indeed  constitute  a  New  Engine  which 
helps  the  mind  as  tools  do  help  the  hand,  it  is  through  the  en- 
richment of  the  materials  of  sense  by  the  operation  of  this  New 
Engine  that  the  elaborate  interpretations  of  the  physical  sciences 
are  made  possible,  and  the  study  of  elementary  physics  is  intended 
to  lead  to  the  realization  of  this  New  Engine  :  {a)  By  the  building 
up  in  the  mind,  of  the  logical  structure  of  the  physical  sciences ; 
(J?)  by  training  in  the  making  of  measurements  and  in  the  per- 
formance of  ordered  operations,  and  {c)  by  exercises  in  the  appli- 
cation of  these  things  to  the  actual  phenomena  of  physics  and 
chemistry  at  every  step  and  all  of  the  time  with  every  possible 
variation. 

That,  surely,  is  a  sufficiently  uncompromising  program.  In- 
deed, many  raise  the  objection  that  a  rigorous  presentation  of 
the  structure  of  physics,  logical  and  mechanical,  is  highly  un- 
satisfactory and  uninstructiv'C,  and  of  course  this  is  true  if  the 
physical  facts  themselves  are  lost  to  view ;  but  any  student  who 


INTRODUCTION.  1 7 

indulges  a  fancied  interest  in  the  "  results  "  of  science,  and  who, 
becoming  absorbed,  for  example,  in  a  purely  descriptive  treatise 
on  recent  researches  on  '*  light  pressure  "  and  the  cause  of  comets' 
tails,  holds  his  imagination  unresponsive  to  a  discussion  of  velocity 
and  acceleration  such  as  that  given  in  the  next  chapter,  any  stu- 
dent who  does  this,  I  say,  should  be  treated  honestly  and  placed 
under  the  instruction  of  Jules  Verne,  where  he  need  not  trouble 
himself  about  foundations,  but  may  follow  his  teacher  pleasantly 
on  a  care -free  trip  to  the  moon,  or  with  easy  improvidence  embark 
on  a  voyage  of  twenty  thousand  leagues  under  the  sea.  There 
are  too  many  people  who  fancy  that  they  have  an  interest  in  the 
"results  "  of  science  and  who,  poor  fools,  invest  in  Keeley  Motors 
and  Sea  Gold  Companies  because,  forsooth,  the  desired  result  is 
so  clearly  evident. 

The  greatest  fault  in  an  elementary  treatise  on  physics  is  ob- 
scurity in  that  region  which  lies  between  raw  unformulated  nature 
on  the  one  hand  and  the  highly  elaborated  ideas  and  conceptions 
of  physical  theory  on  the  other  hand.  *'  Our  method,"  says 
Bacon,  "  is  continually  to  dwell  among  things  soberly,  without 
abstracting  or  setting  the  mind  farther  from  them  than  makes 
their  images  meet,"  and  '*  the  capital  precept  for  the  whole  under- 
taking is  that  the  eye  of  the  mind  be  never  taken  off  from  things 
themselves,  but  receive  their  images  as  they  truly  are,  and  God 
forbid  that  we  should  offer  the  dreams  of  fancy  for  a  model  of 
the  world." 


CHAPTER    11. 

MEASUREMENT   OF   LENGTH,    ANGLE,    MASS   AND   TIME. 

1.  Units  of  length.  —  The  meter  is  the  distance,  at  the  tempera- 
ture of  melting  ice,  between  two  Hnes  on  a  certain  platinum-iridium 
bar  which  is  preserved  in  the  vaults  of  the  International  Bureau 
of  Weights  and  Measures  near  Paris.  Very  accurate  copies  *  of 
this  bar  are  deposited  in  Washington  and  in  London,  and  the  legal 
units  of  length  in  all  countries  are  now  defined  in  terms  of  the 
meter. 

It  was  intended  originally  that  the  meter  should  be  equal  to 
one  ten-millionth  part  of  the  distance  from  the  equator  to  the 
poles  of  the  earth,  but  copies  of  the  meter  can  be  made  with 
much  greater  accuracy  and  with  incomparably  greater  ease  by- 
direct  comparison  with  the  standard  meter  bar  in  Paris  than  by 
comparison  with  the  earth's  quadrant,  and  therefore  the  definition 
of  the  meter  as  a  ten-millionth  of  the  earth's  quadrant  is  entirely 
illusory. 

The  yard  is  now  defined  as  ffff  of  a  meter,  f 

2.  Measurement  of  length :  scale  and  vernier.  —  Lengths  are 
commonly  measured  by  means  of  divided  scales.  The  measure- 
ment is  accomplished  by  counting  the  number  of  scale  divisions 
included  in  the  length  to  be  measured.  Fractions  of  a  division 
may  be  estimated  by  the  eye  or  determined  by  means  of  a  device 
called  a  vernier.     The  general  principle  of  the  vernier  is  as  fol- 

*  These  copies  are  called  the  international  prototypes  of  the  meter.  See  Nature, 
Vol.  51,  p.  420,  February  28,  1895. 

f  On  April  5,  1893,  ^  decision  was  reached  by  the  United  States  Superintendent 
of  Weights  and  Measures,  with  the  approval  of  the  Secretary  of  the  Treasury,  that 
the  meter  and  the  kilogram  would  be  regarded  as  the  fundamental  standards  not  only 
for  metric  units  but  also  for  the  customary  units  of  length  and  mass.  See  a  History 
of  the  Standard  "Weights  and  Measures  of  the  United  States  by  Louis  A.  Fischer, 
Vol.  L,  pp.  365-381,  Bulletin  of  the  Bureau  of  Standards  (United  States  Depart- 
ment of  Commerce  and  Labor). 

18 


MEASUREMENT   OF    MASS    AND    TIME. 


19 


lows :  The  divided  scale  is  represented  by  S,  Fig.  i .  Let  us 
call  the  divisions  on  this  scale  millimeters  for  brevity.  A  short 
auxiliary  scale  V,  the  vernier,  is  (n  —  i)  millimeters  long  and  it 
is  divided  into  7i  equal  parts.  The  diagram,  Fig.  i,  is  constructed 
for  ?/  =  10.  Let  the  space  /  be  the  fraction  of  a  millimeter  to  be 
determined,  and  let  it  be  equal  to  a/n  of  a  millimeter  ;  the  space 
£•  is  1/71  mm.  shorter,  the  space  A  is  2/71  mm.  shorter,  and  so 
on  ;  so  that  the  a^^  mark  on  the  vernier  is  coincident  with  a  mark 
on  the  scale.  Thus  a  is  determined.  The  number  of  the  mark 
on  the  vernier  which  is  coincident  with  a  mark  on  the  scale  is  the 


^1^1^  [^[1-  g=-fe|inife  W- 


1 5-;.  ^  =^/T2o.  If  Ji  ==---  '25; 


30 


m 


s 

Fig.  1. 

numerator  and  the  number  of  divisions  on  the  vernier  is  the 
deno77iinator  of  the  fraction  which  expresses  the  space  /  in  terms 
of  a  scale  division.  The  position  of  the  zero  mark  of  the  vernier 
on  the  scale  is  called  the  readittg  of  the  ver7iier.  Thus  the  read- 
ing of  the  vernier  in  Fig.  i  is  20.7  mm. 

To  measure  a  given  length,  any  chosen  point  on  the  vernier  is 
made  to  coincide  with  one  end  of  the  length,  and  the  vernier  read- 
ing is  taken.  The  vernier  is  then  moved  until  the  same  point  of 
the  vernier  coincides  with  the  other  end  of  the  length,  and  the 
vernier  reading  is  again  taken.  The  difference  of  these  two  ver- 
nier readings  is  the  distance  the  vernier  has  been  moved  and  it  is 
equal  to  the  given  length. 

The  vernier  is  frequently  used  to  determine  fractions  of  divi- 
sions on  divided  circles. 

3.  The  dividing  engine  is  a  machine  for  the  manufacture  of 
divided  scales  and  for  the  precise  measurement  of  length.  An 
accurate  *  horizontal  screw,  having  a  divided  circular  head  for 

*  An  interesting  description  of  the  method  of  making  an  accurate  screw  is  given  in 
the  Encyclopedia  Britannica,  9th  edition,  article  screw. 


20 


ELEMENTS   OF   MECHANICS. 


indicating  fractions  of  a  turn,  engages  a  nut  which  pushes  a  slid- 
ing carriage  along  a  track  on  a  heavy  metal  bed-plate  which  pro- 
jects to  one  side  of  the  carriage  as  a  platform,  as  shown  in  Fig. 
2.  On  the  carriage  are  mounted  a  graving  tool  and  a  reading 
microscope  (the  dividing  engine  shown  in  Fig.  2  is  intended  only 
for  the  making  of  divided  scales,  its  screw  is  standardized  by  the 
manufacturer,  and  it  is  not  intended  for  use  in  measuring  the 


MEASUREMENT    OF    MASS    AND    TIME.  21 

length  of  an  object).  To  standardize  the  screw  a  meter  bar  is 
placed  on  the  platform,  the  screw  is  turned  until  the  microscope 
sights  at  one  end  of  the  meter,  and  then  the  exact  number  of 
turns,  a,  required  to  move  the  microscope  to  the  other  end  of 
the  meter  is  counted,  fractions  of  turns  being  estimated  by  the 
divided  circular  head. 

To  measure  the  length  of  any  object,  the  object  is  placed  on 
the  platform,  the  screw  is  turned  until  the  microscope  sights  at 
one  end  of  the  object,  and  then  the  exact  number  of  turns,  ^, 
required  to  move  the  microscope  to  the  other  end  of  the  object 
is  counted.  The  length  of  the  object  is  then  known  to  be  b\a 
of  a  meter. 

To  manufacture  a  divided  scale,  a  blank  bar  is  placed  upon  the 
platform  of  the  dividing  engine,  the  screw  is  turned  until  the 
graving  tool  is  conveniently  near  to  one  end  of  the  bar,  and  a 
mark  is  made ;  the  screw  is  then  turned,  aln  turns,  and  an- 
other mark  is  made,  and  so  on,  thus  dividing  the  bar  into  ;/ths 
of  a  meter. 

4.  Units  of  angle.  The  angle  all  the  way  around  a  point, 
that  is,  the  angle  which  is  represented  by  the  entire  circumfer- 
ence of  a  circle,  is  a  natural  unit  of  angle,  and  it  is  not  neces- 
sary to  preserve  'a  material  standard  of  the  unit  angle.  The  unit 
of  angle  which  is  universally  used  for  purposes  of  measurement 
is  the  degree,  it  is  equal  to  -^\-^  of  the  angle  which  is  repre- 
sented by  the  entire  circumference  of  a  circle.  In  many  calcu- 
lations, however,  it  is  convenient  to  express  an  angle  as  follows  : 
Imagine  a  circle  of  radius  r  drawn  with  its  center  at  the  apex  of 
an  angle,  and  let  a  be  the  length  of  the  arc  of  the  circle  which  is 
included  between  the  boundaries  of  the  angle  ;  then  the  ratio 
a\r  has  a  fixed  value  for  a  given  angle,  and  the  value  of  this 
ratio  is  frequently  used  as  a  numerical  measure  of  the  angle. 
The  unit  angle  in  this  system  is  the  angle  of  which  the  length  of 
the  subtending  arc  is  equal  to  the  radius  and  it  is  called  the  radian. 

Measurement  of  angle.  —  In  many  instruments,  angles  are 
measured  by  means  of  the  divided  circle.     The  divided  circle  is 


22 


ELEMENTS   OF   MECHANICS. 


placed  with  its  center  at  the  apex  of  the  angle,  and  the  angle  is 
measured  by  counting  the  number  of  circle  divisions  between  the 
lines  which  determine  the  angle.  These  lines  are  established  by 
a  pair  of  sights  fixed  to  an  arm  called  an  alidade.  This  alidade 
rests  flat  on  the  divided  circle,  turns  on  a  pivot  at  the  center  of 
the  circle,  and  carries  usually  two  verniers,  one  at  each  end. 

Indirect  ?neasure??ieni  of  length  and  angle : 
triangulation.  —  Consider  a  triangle  ABC^  Fig.  3, 
of  which  the  side  b,  only,  is  accessible.  The 
lengths  of  the  sides  a  and  c  and  the  angle  B  may  be 
calculated  by  trigonometry  when  the  side  b  and  the 
angles  A  and  C  have  been  measured.  This  method 
for  determining  inaccessible  lengths  is  called  tri- 
angulation. It  is  much  used  in  surveying  and  in 
astronomy. 

Poggendorff'' s  method  for  measuring  angle. — 
In  many  instruments  it  is  necessary  to  measure  the 
angle  through  which  a  suspended  body  is  turned. 
For  this  purpose  a  small  mirror,  mm,  Fig.  4,  is 
fastened  to  the  suspended  body  so  as  to  turn  with  it.  A  straight  scale,  ss,  is  placed  at  a 
measured  distance,  d,  in  front  of  and  parallel  to  the  mirror.  A  telescope,  which 
establishes  a  sight  line,  is  placed  so  that  the  scale  is  seen  through  it  in  the  mirror. 


Fig.  3. 


Fig.  4. 


the  sight  line  being  perpendicular  to  the  scale.     The  reading,  a,  of  this  sight  line  on 
the  scale  is  taken  ;  the  mirror  then  turns  through  the  angle  Q  into  the  position  m^m't 


MEASUREMENT   OF   MASS   AND   TIME.  23 

and  the  scale  reading,  b,  is  again  taken.     The  sight  line  is  deflected  through  the 
angle  2.B,  so  that  (a  —  d)/d=ta.n  20,  from  which  the  angle  0  may  be  calculated. 

5.  Units  of  area.  —  The  unit  of  area  is  defined  as  the  area  of 
a  square  *  of  which  the  side  is  of  unit  length. 

Measurement  of  area.  —  Area  is  determined  fundamentally  by- 
calculation  from  measured  linear  dimensions. 

The  planimeter  is  an  instrument  for  measuring  irregular  plane 
areas ;  strictly,  for  determining  the  ratio  of  two  such  areas,  for 
the  instrument  is  standardized  by  measuring  with  it  a  regular  figure 
of  which  the  area  is  known  by  calculation  from  measured  linear 
dimensions. 

Theory  of  the  planimeter. — Consider  a  line  AB,  Fig.  5,  of  length  /,  moving  in 
any  manner  in  the  plane  of  the  paper.  The  motion  of  the  line  may  at  each  instant  be 
considered  as  compounded  f  of  a  motion  of  translation  and  a  motion  of  rotation,  with 
angular  velocity  dd  j  dt,  about  an  arbitrary  point  /  distant  D  from  the  center  of  the 
line.  The  area  swept  by  this  moving  line  is  considered  positive  when  the  line 
sweeps  over  it  from  left  to  right  to  an  observer  looking  from  A  towards  B.  Let  v  be 
the  resolved  part,  perpendicutar  to  the  line,  of  its  velocity  of  translation.  The  line 
sweeps  over  area  at  the  rate  Iv,  because  of  its  motion  of  translation,  and  at  the  rate 
ID  ■  dd  Idty  because  of  its  motion  of  rotation,  so  that  the  total  rate  at  which  the  line 
sweeps  area  is  at  each  instant : 

Let  a  wheel,  radius  r,  mounted  at  p,  with  its  axis  parallel  to  AB,  be  allowed  to 
roll  on  the  paper  as  the  line  moves,  and  let  d'\\)ldt  be  the  angular  velocity  of  rolling 
of  the  wheel.     Then  z/  =  r  •  dipjdt,  and  equation  (i)  becomes 

orf 

A  =  lryl)-\-  IDd  (iii) 

in  which  A  is  the  total  area  swept  by  the  line  during  the  time  that  the  wheel  has 
turned  through  the  angle  ^p  and  the  line  has  turned  about  /  through  the  angle  6. 

*The  circular  mil,  much  used  by  electricians  as  a  unit  area,  is  the  area  of  a  circle 
one  mil  (xtjW  '^^^^)  ^^  diameter.  The  area  of  any  circle  in  circular  mils  is  equal  to 
the  square  of  its  diameter  in  mils. 

I  See  Article  77. 

X  See  Article  20. 


24 


ELEMENTS   OF   MECHANICS. 


If  the  line  comes  back  to  its  initial  position,  or  parallel  thereto,  so  that  6  is  equal 
to  zero,*  then  equation  (iii)  becomes 


Ir^ 


(iv) 


That  is,  the  total  area  swept  by  AB  is  proportional  to  the  angle  ip  turned  by  the 
rolling  wheel,  and  /Ag  circutnference  of  the  ivheel  may  be  so  divided  as  to  read  ai-eas 
directly. 

Let  one  end  of  AB^  Fig.  6,  be  constrained  to  move  along  a  branch  ^C  of  any 
curve,  while  the  other  end  passes  once  around  a  closed  line  D.  Any  area  outside  of 
D  which  is  swept  over  by  the  line   AB   at  all  is  swept  as  many  times  to  the  right  as 


Fig.  5. 


Fig.  6. 


to  the  left,  and  all  parts  of  D  are  swept  once  more  to  the  right  than  to  the  left,  so 
that  the  total  area  swept  by  AB  is  equal  to  the  area  of  D.  In  its  simplest  form  the 
planimeter  consists  of  an  arm  AB  with  a  rolling  wheel.  The  end  A  is  constrained 
to  move  on  the  arc  of  a  circle,  being  hinged  to  one  end  of  an  auxiliary  arm,  the 
other  end  of  which  is  fixed  by  a  pivot. 

6.  Units  of  volume.  —  The  unit  of  volume  is  defined  as  the 
volume  of  a  cube  of  which  the  edge  is  of  unit  length. 

Measurerne^it  of  volujne.  —  Volume  is  determined  fundamen- 
tally by  calculation  from  measured  linear  dimensions.  Thus  the 
volume  of  a  rectangular  parallelopiped  is  equal  to  the  product 
of  its  length,  breadth,  and  height. 

Volumes  of  liquid  and  of  grain  are  usually  measured  by  means 
of  a  vessel  of  known  volume.  The  graduate  is  a  vessel  upon 
the  side  of  which  is  a  scale  from  which  the  volume  of  a  portion 
of  liquid  may  be  read  off;    it  is  much  used  by  druggists  and 

*  The  line  may  come  back  to  its  initial  position  so  that  d  equals  2«7r,  where  n  is 
any  whole  number. 


MEASUREMENT   OF   MASS   AND   TIME.  25 

by  chemists.  A  method  for  determining  volume  by  weighing  is 
described  under  density. 

7.  Mass.  —  Everyone  is  famihar  with  the  measurement  of 
material  by  volume  and  by  weighty  but  everyone  does  not  distin- 
guish between  the  two  methods  in  common  use  for  measuring  by 
weight,  namely,  the  method  in  which  the  spring  scale  is  used  and 
the  method  in  which  the  balance  scale  is  used.  The  spring  scale 
measures  the  force  with  which  the  earth  pulls  a  body,  and  the 
weight  of  a  given  body  as  indicated  by  a  spring  scale  is  greater 
or  less  according  as  the  pull  of  the  earth  for  the  given  body  is 
greater  or  less  ;  the  indication  of  the  balance  scale,  on  the  other 
hand,  does  not  vary  with  the  gravity -pull  of  the  earth,  inasmuch 
as  the  gravity-pull  on  the  weights  and  the  gravity-pull  on  the 
weighed  body  both  change  together,  the  indications  of  a  balance 
scale  are  therefore  independent  of  the  value  of  gravity.  The 
result  of  the  operation  of  weighing  by  a  bala^ice  scale  is  called  the 
mass  of  a  body.^ 

Considering  that  weighing  is  nearly  always  done  by  the  bal- 
ance scale,  it  is  evident  that  what  is  popularly  called  the  weight 
of  a  body  is  what  scientific  men  call  the  mass  of  the  body,  and  it 
is  important  to  remember  that  the  force  with  which  the  earth 
pulls  a  body  is  called  the  weight  of  the  body  by  scientific  men. 
The  verb  to  weigh  means  nearly  always  the  determination  of  the 
mass  of  a  body  by  means  of  the  balance  scale. 

*  We  might  agree  to  consider  the  mass  of  one  body  to  be  twice  as  great  as  the  mass 
of  another  body  when  a  given  force  will  produce  half  as  much  velocity  per  second 
when  it  acts  upon  the  first  body  as  it  will  when  it  acts  upon  the  second  body  ;  but  the 
proper  definition  of  a  quantity  is  the  definition  which  corresponds  to  the  fundamental 
method  which  is  actually  used  in  measuring  that  quantity.  No  one  ever  thinks  of 
measuring  out  a  ton  of  coal  by  loading  it  upon  a  "  frictionless  "  car  and  finding  how 
many  times  less  velocity  is  imparted  to  it  in  a  second  by  a  given  force  than  would  be 
imparted  to  a  standard  pound  by  the  same  force  !  Compare  this  as  an  actual  opera- 
tion with  the  measuring  of  coal  by  the  ordinary  platform  scale,  using  standard  pounds 
and  fractions  of  a  pound  as  counterpoises  !  It  is  all  very  well  to  talk  about  defining 
the  mass  of  a  body  in  accordance  with  the  above  utterly  impracticable  method  of 
measuring  its  mass,  but  sensible  men  always  define  things  in  physics  in  the  way  they 
do  them. 


26  ELEMENTS    OF   MECHANICS. 

If  the  force  with  which  the  earth  pulls  the  unit  of  mass  at  a 
giveji  place  on  the  earth  is  adopted  as  the  unit  of  force  and  called 
the  poimd  of  force  (or  the  grain  of  force),  then  the  weight  of  any 
body  in  pounds  of  force  at  that  place  on  the  earth  will  be  exactly 
equal  to  its  mass  in  pounds,  and,  in  fact,  the  weight  of  a  body  at 
any  other  place  on  the  earth  will  not  differ  from  its  mass  by 
more  than  two  or  three  tenths  of  one  per  cent. 

Units  of  mass.  The  kilogram  is  the  mass  of  a  certain  piece 
of  platinum  which  is  preserved  in  the  vaults  of  the  International 
Bureau  of  Weights  and  Measures,  Very  accurate  copies  *  of  the 
kilogram  are  deposited  in  Washington  and  in  London,  and  the 
legal  units  of  mass  in  all  countries  are  now  defined  in  terms 
of  the  kilogram.  Thus  the  pound  (avoirdupois)  is  defined  as 
1/2.204622  of  a  kilogram. t 

It  was  intended  originally  that  the  kilogram  should  be  equal 
to  the  mass  of  a  cubic  decimeter  (1,000  cubic  centimeters)  of 
water  at  a  temperature  of  4°  C.  and  at  atmospheric  pressure, 
but  the  extreme  difficulty  of  reproducing  accurate  copies  of  the 
kilogram  on  the  basis  of  this  definition  makes  the  definition  not 
only  impracticable  but  illusory. 

Measurement  of  mass.  —  The  analytical  balance  consists  of  a 
delicately  mounted  equal-arm  lever  with  pans  suspended  from  its 
ends.  The  balance  is  used  simply  for  indicating  the  equality  of  the 
masses  of  two  bodies,  that  is,  two  bodies  are  said  to  have  equal 
masses  when  they  balance  each  other  when  suspended  from  the 
ends  of  an  equal-arm  lever. 

The  determination  of  the  mass  of  a  body  by  means  of  the  bal- 
ance depends  upon  the  use  of  a  set  of  weights  which  may  be  com- 
bined in  such  a  way  as  to  match  the  mass  of  the  body.  Such  a 
set  of  weights  may  be  made  by  taking  two  pieces  of  metal  weigh- 
ing together  one  kilogram  and  then  making  them  balance  each 
other  by  cutting  metal  off  from  one  and  adding  the  shavings  to 

*See  Nature,  Vol.  51,  p.  420,  February  28,  1895. 

fSee  Bulletin  of  the  Bureau  of  Standards  (United  States  Department  of  Com- 
merce and  Labor),  Vol.  I.,  pp.  365-381. 


MEASUREMENT   OF   MASS   AND    TIME.  2/ 

the  other,  thus  giving  a  half-kilogram  weight.  Then  a  quarter- 
kilogram  weight  may  be  made  in  the  same  way  and  so  on.  A  set 
of  weights  more  convenient  in  use  is  a  set  which  contains  a  five,  a 
two,  and  two  ones  of  each  —  units,  tens,  hundreds,  etc.,  of  grams. 

8.  Density.  Every  one  knows  that  lead  is  heavier  than  cork, 
and  every  one  feels  instantly  that  there  is  some  hitch  in  the  ques- 
tion **  Which  is  heavier,  a  pound  of  lead  or  a  pound  of  cork." 
The  word  heaviness  has,  in  fact,  a  double  meaning.  A  pound  of 
lead  is  not  heavier  than  a  pound  of  cork,  because  to  specify  a 
pound  in  each  case  is  to  imply  that  both  have  been  weighed  and 
that  the  result  is  the  same  for  each,  namely,  one  pound  of  lead 
and  one  pound  of  cork ;  but  lead  is  heavier  than  cork  in  the 
sense  that  a  piece  of  lead  weighs  more  than  an  equal  bulk  of  cork. 
The  word  density  is  used  to  designate  this  idea  of  heaviness  as  an 
inherent  property  of  a  substance.  Thus,  lead  has  greater  density 
than  cork. 

The  density  of  a  substance  is  its  mass  per  unit  of  volume,  that 
is,  it  is  the  mass  of  a  body  divided  by  the  volume  of  the  body. 
The  density  of  a  substance  may  be  specified  in  terms  of  any  units 
of  mass  and  volume.  Thus  the  density  of  a  Hquid  such  as  oil  is 
usually  specified  by  commercial  men  as  so  many  pounds  per 
gallon  ;  the  density  of  stone  is  usually  specified  by  a  building 
contractor  as  so  many  pounds  per  cubic  foot,  and  so  on.  In  this 
treatise  density  will  be  expressed  in  grams  per  cubic  centimeter. 

The  specific  gravity  of  a  substance  at  a  given  temperature  is 
the  ratio  of  the  density  of  the  substance  to  the  density  of  water 
at  the  same  temperature,  that  is  to  say,  the  specific  gravity  of  a 
substance  is  its  density  expressed  in  terms  of  the  density  of  water 
as  unity. 

Measurement  of  density.  —  The  density  of  a  substance  is  deter- 
mined fundamentally  by  weighing  a  measured  volume  of  the  sub- 
stance. Thus  the  density  of  water  has  been  very  carefully  deter- 
mined at  the  International  Bureau  of  Weights  and  Measures  *  as 

*W.  Marek,  Trav.  et  Mhn.  du  Bureau  internat.  des  Poids  et  Mes.,  Ill,  D  8i, 
1884. 


28     ■  ELEMENTS   OF   MECHANICS. 

follows  :  The  volume  of  an  accurately  cut  glass  cube  is  calculated 
from  its  measured  linear  dimensions  and  the  cube  is  weighed 
in  air  and  then  suspended  under  water  and  weighed  again.  The 
difference,  duly  corrected  for  buoyancy  of  air,  is  the  mass  of  a 
volume  of  water  equal  to  the  volume  of  the  cube.  The  density 
of  water  at  a  given  temperature  being  thus  determined,  its  density 
at  other  temperatures  is  found  by  the  method  of  Regnault  which 
is  described  in  the  chapter  on  thermometry. 

Measuremeitt  of  specific  gravity.  —  The  specific  gravity  of  a 
substance  is  determined  by  weighing  equal  volumes  of  the  sub- 
stance and  of  water.  The  simplest  case  is  in  the  use  of  the 
specific  gravity  bottle  for  determining  the  specific  gravity  of  a 
liquid.  The  bottle  is  weighed  empty,  then  it  is  weighed  when 
filled  with  the  given  liquid,  and  then  it  is  weighed  when  filled 
with  water.  Other  methods  for  determining  specific  gravity  are 
discussed  in  the  chapter  on  hydrostatics.  When  the  specific 
gravity  of  a  substance  has  been  found  at  a  given  temperature, 
the  density  of  the  substance  at  the  given  temperature  may  be 
found  by  multiplying  the  specific  gravity  of  the  substance  by  the 
known  density  of  water  at  that  temperature.  See  table  dejtsity 
of  water  in  the  chapter  on  thermometry. 

Gravimetric  method  for  measuring  volume.  —  The  volume  of  a 
vessel  at  a  given  temperature  may  be  determined  with  great  ac- 
curacy by  weighing  the  vessel  empty,  and  then  weighing  it  when 
filled  with  water  or  mercury  at  the  given  temperature.  The 
density  of  the  water  or  mercury  being  known,  the  volume  of  the 
vessel  is  easily  calculated  from  the  net  weight  (mass)  of  the  water 
or  mercury.  The  measuring  vessels  used  by  chemists  are  stan- 
dardized in  this  way. 

9,  Time.  —  The  mean  solar  day  is  the  natural  unit  of  time, 
and  the  second,  which  is  the  accepted  unit  of  time  in  many  physi- 
cal measurements,  is  the  86,400th  part  of  a  mean  solar  day. 

Measurement  of  time.  —  Any  movement  of  a  body  which  re- 
peats itself  in  equal  intervals  of  time  is  called  periodic  motion ; 
single  movements  are  called  vibrations.     All  methods,  with  un- 


MEASUREMENT   OF   MASS   AND   TIME.  29 

important  exceptions,  for  measuring  time  depend  upon  periodic 
motion.  A  vibrating  pendulum  is  the  most  familiar  example. 
The  7iuinbery  a,  of  vibrations  in  a  day,  ajtd  the  number,  b,  in  the 
interval  to  be  measured,  are  counted.  The  interval  is  then  known 
to  be  equal  to  bfa  of  a  day.  A  clock  is  simply  a  machine  for 
maintaining  and  counting  the  vibrations  of  a  pendulum.  In 
portable  clocks  a  balance  ivheel  takes  the  place  of  a  pendulum. 

Anything  which  affects  the  time  of  vibration  of  a  pendulum 
leads  to  erroneous  values  for  time  intervals  as  measured  by  a 
clock.  The  time  of  vibration  of  a  pendulum  is  affected  {a)  by 
temperature,  on  account  of  increase  of  length  of  the  pendulum 
with  temperature ;  {b)  by  variations  of  atmospheric  pressure,  on 
account,  mainly,  of  variation  of  buoyant  force  of  air  with  pressure  ; 
(r)  by  variation  in  amplitude  of  vibration  ;  and  (d^  by  irregularities 
in  the  manner  in  which  impulses  are  imparted  to  the  pendulum  by 
the  clockwork.  The  influence  of  temperature  is  avoided  by  using 
what  are  called  compensated  pendulums,  which  do  not  change 
their  effective  length  with  temperature.  The  variations  due  to 
{/)  are  in  part  obviated  by  providing  constant  driving  power, 
which  requires  the  gears  and  escapement  to  be  of  fine  workman- 
ship. The  variations  due  to  {d')  are  obviated  by  using  what  are 
called  dead  beat  escapements,  which  impart  their  impulse  to  the 
pendulum  at  the  instant  when  it  passes  through  the  vertical 
position.* 

10.  The  chronograph.  —  The  determination  of  a  time  interval 
by  means  of  a  clock  requires  the  clock  reading  to  be  taken  at  the 
beginning  and  at  the  end  of  the  interval.  Practice  enables  an 
observer  to  take  the  clock  reading  at  the  instant  of  a  given  signal 
accurately  to  a  tenth  of  a  second,  with  an  approximately  constant 
"  personal "  error  which  does  not  greatly  affect  the  value  of  the 
interval.  In  the  practice  of  this  method,  which  is  called  the  eye 
and  ear  method,  the  observer  looks  for  the  signal  and  listens  to 
the  beats  of  the  clock. 

*  For  discussion  of  errors,  and  description  of  escapements,  see  Encyclopedia  Britan- 
nica,  9th  ed.,  article  Clock.  For  description  of  compensated  chronometer  balance  and 
chronometer  escapement,  see  Lockyer's  book  entitled  Star-gazing,  pp.  175  to  210. 


30. 


ELEMENTS   OF   MECHANICS. 


The  chronograph  is  an  instrument  for  enabling  clock  readings 
to  be  taken  with  greater  ease  and  accuracy  than  is  possible  by 
''eye  and  ear."  A  pen  traces  a  Hne  upon  a  uniformly  moving 
strip  of  paper.  This  pen  is  fixed  to  the  armature  of  an  electro- 
magnet, which  is  excited  at  each  beat  of  the  clock  by  an  electric 
current   controlled  by  a  contact  device  actuated  by  the  clock 


MEASUREMENT   OF   MASS   AND   TIME.  3 1 

pendulum.  A  kink  is  thus  made  in  the  traced  line  at  each  beat 
of  the  pendulum.  At  the  instant  for  which  the  clock  reading  is 
desired,  the  electro-magnet  is  momentarily  excited  by  pressing  a 
key  which  closes  an  auxiliary  electric  circuit,  thus  making  an 
extra  kink  in  the  traced  line,  and  the  clock  reading  is  determined 
by  measuring  off  the  position  of  this  extra  kink  among  the  kinks 
produced  by  the  beats  of  the  pendulum. 

The  above  description  applies  to  the  essential  features  of  the 
chronograph.  The  form  of  the  instrument  as  ordinarily  used  for 
accurate  time  observations  is  shown  in  Fig.  7.  A  large  sheet  of 
paper  is  wrapped  around  a  cylinder  which  is  rotated  at  a  uniform 
speed  by  clock  work.  The  tracing  pen  and  electro-magnet  are 
mounted  on  a  sliding  carriage  which  is  slowly  moved  parallel  to 


ZF^C 


1  f. 

„ 

„ 

„ 

„ 

r. 

n 

, JX 

„ 

^ 

„ 

„ 

r? 

n 

pi 

Fig.  8. 

the  axis  of  the  rotating  cylinder  by  a  screw  which  is  driven  by 
the  same  clock-work  that  drives  the  cylinder.  The  pen  thus 
traces  a  helical  line  on  the  paper-covered  cylinder,  so  that  the 
large  sheet  of  paper  is  equivalent  to  a  very  long  narrow  strip. 

Figure  8  shows  a  reduced  facsimile  of  a  portion  of  the  paper 
sheet  upon  which  a  chronographic  record  has  been  made.  The 
regularly  spaced  kinks  are  those  produced  by  the  beats  of  the 
clock,  and  the  kinks  marked  a^  b,  c,  etc.,  are  those  produced  by 
depressing  the  key. 

Problems. 

1.  A  scale  has  half  inch  divisions.  A  vernier  to  be  used  with 
this  scale  is  7J  inches  long  and  it  is  divided  into  sixteen  equal 
parts.     To  what  fractional  part  of  an  inch  does  the  vernier  read  ? 

2.  The  divisions  on  the  circle  of  a  surveyor's  transit  are  J  of  a 


32  .  ELEMENTS   OF   MECHANICS. 

degree.  A  vernier  is  to  be  made  for  use  with  this  circle  and  to 
read  to  lo''.  What  is  the  length  of  the  vernier  and  what  is  the 
number  of  divisions  upon  it  ? 

3.  Reduce  an  angle  of  20°  to  radians.  Reduce  an  angle  of 
one  radian  to  degrees. 

4.  The  density  of  alcohol  is  6. 3  5  pounds  per  gallon.  What  is 
its  density  in  pounds  per  cubic  inch  ?  What  is  its  density  in 
grams  per  cubic  centimeter? 

Note.  — One  gallon  equals  231  cubic  inches.  One  inch  equals  2.54  centimeters. 
One  pound  equals  453.6  grams. 

5.  The  density  of  iron  is  j .Z  grams  per  cubic  centimeter.  What 
is  its  density  in  pounds  per  cubic  inch  ?  What  is  the  mass  in 
pounds  of  an  iron  bar  30  feet  long  and  5  J  square  inches  sectional 
area  ? 

6.  A  bottle  weighs  50.62  grams  empty,  and  288.93  grams  full 
of  water  at  21°  C.  What  is  the  cubic  contents  of  the  bottle  ? 
The  bottle  full  of  oil  at  21°  C.  weighs  239.2  grams.  What  is 
the  specific  gravity  of  the  oil  ?  What  is  the  density  of  the  oil  ? 
Neglect  the  effect  of  buoyant  force  of  the  air. 

Note.  —  See  chapter  on  thermometry  for  table  of  densities  of  water  at  various  tem- 
peratures. 

7.  A  clock  with  a  pendulum  which  is  supposed  to  beat  seconds 
reads  12'',  o"",  I6^8  at  mean  noon  on  one  day,  and  it  reads  I2^ 
I™,  2^3  at  mean  noon  ten  days  later.  What  number  of  beats 
does  the  pendulum  of  the  clock  make  in  a  mean  solar  day  ?  What 
is  the  true  value  in  hours",  minutes,  and  seconds  of  an  interval  of 
time  at  the  beginning  of  which  the  above  clock  reads,  5*",  6™, 
io'.2,  and  at  the  end  of  which  the  clock  reads  8^  33",  56'.7? 


CHAPTER   III. 
PHYSICAL   ARITHMETIC. 

11.  Measures ;  units.  —  In  the  expression  of  a  physical  quantity- 
two  factors  always  occur,  a  numerical  factor  and  a  unit.  The 
numerical  factor  is  called  the  measure  of  the  quantity.  Thus  a 
certain  length  is  65  centimeters,  a  certain  time  interval  is  250 
seconds,  a  certain  electric  current  is  25  amperes,  a  certain  elec- 
tromotive force  is  1 1  o  volts. 

It  is  a  great  help  towards  a  clear  understanding  of  physical 
calculations  to  consider  that  both  units  arid  measures  are  involved 
in  a  product  of  two  physical  quantities  or  in  a  quotient  of  two 
physical  quantities.  Thus  a  rectangle  is  5  centimeters  wide  and 
10  centimeters  long  ;  and  its  area  is  5  centimeters  times  10  centi- 
meters, which  is  equal  to  50  square  centimeters.  A  cylinder  is 
10  centimeters  long  and  the  area  of  one  of  its  ends  is  25  square 
centimeters;  and  its  volume  is  10  centimeters  times  25  square 
centimeters,  which  is  equal  to  250  cubic  centimeters.  A  train 
travels  500  feet  in  10  seconds  and  its  average  velocity  during  the 
time  is  500  feet  divided  by  10  seconds  which  is  equal  to  50  feet 
per  second.  A  body  is  dragged  through  a  distance  of  1 5  feet  by 
a  force  of  10  pounds  and  the  amount  of  work  done  is  15  feet 
times  10  pounds,  which  is  equal  to  \<^o  foot-pounds.  The  word 
per  connecting  the  names  of  two  units  indicates  that  the  unit  fol- 
lowing is  a  divisor,  thus  a  velocity  of  50  feet  per  second  may  be 
and  often  is  written  50  feet /second.  A  hyphen  connecting  the 
names  of  two  units  indicates  a  product  of  the  units ;  products  and 
quotients  of  units  arrived  at  in  this  way  are  always  new  physical 
units.  Thus  the  foot  per  second  is  a  unit  of  velocity,  the  foot- 
pound is  a  unit  of  work. 

It  is  important  to  carry  the  units  through  with  every  numerical 
calculation,  the  arithmetical  operations  among  the  various  units 

3  33 


34  ELEMENTS   OF   MECHANICS. 

being  indicated  algebraically.  Whe?i  this  is  done  there  can  be  no 
arnbiguity  as  to  the  meaning  of  the  result^  and  when  this  is  not 
done  the  result  has^  strictly  speakings  no  physical  meaning  at  all. 

Although  the  unit  in  terms  of  which  a  result  is  expressed  is 
known  *  when  the  units  are  carried  through  a  numerical  calcula- 
tion, it  frequently  happens  that  the  unit  is  so  entirely  novel  that 
it  might  almost  as  well  be  unknown.  Thus  the  rule  for  finding 
the  area  of  a  rectangle  by  taking  the  product  of  length  and  breadth 
is  entirely  general,  no  matter  what  units  of  length  are  used,  and 
the  area  of  a  rectangle  2  meters  long  and  50  centimeters  wide  is 
equal  to  2  meters  x  50  centimeters  or  100  meter-centimeters. 
Now  the  meter-centimeter  is  a  unit  of  area  equal  to  the  area  of  a 
rectangle  one  meter  long  and  one  centimeter  wide  and  it  is  so  en- 
tirely unfamiliar  as  a  unit  of  area  among  men  engaged  in  practical 
work  that  one  might  almost  as  well  not  know  the  value  of  an 
area  at  all  as  to  have  it  given  in  terms  of  such  a  unit.  It  is,  for 
this  reason,  nearly  always  necessary  to  reduce  the  data  of  a  prob- 
lem to  certain  accepted  units  before  these  data  can  be  used  intelli- 
gibly in  numerical  calculations. 

12.  Units,  fundamental  and  derived.  — 11\\.^  fundamental  physi- 
cal units  are  those  which  are  fixed  by  arbitrary  preserved  stand- 
ards. Thus  the  unit  of  length  is  preserved  as  a  platinum  bar  in 
Paris,  the  unit  of  mass  is  preserved  by  a  piece  of  platinum  in 
Paris,  and  the  second  is  naturally  preserved  in  the  constancy  of 
speed  of  rotation  of  the  earth. 

Derived  physical  tmiis  are  those  which  are  defined  in  terms  of 
the  fundamental  units  and  of  which  no  material  standard  need  be 
preserved.  Thus  the  unit  of  area  is  defined  as  the  area  of  a 
square  of  which  each  side  is  a  unit  of  length,  and  there  is  no 
need  of  preserving  a  material  standard  of  the  unit  of  area.  The 
unit  of  velocity  is  defined  as  unit  distance  traveled  per  second, 
and  there  is  no  need  of  preserving  a  material  standard  of  the  unit 
of  velocity,  indeed,  it  would  be  impracticable  to  preserve  a 
velocity. 

*  See  Art.  14,  on  dimensions  of  derived  units. 


PHYSICAL   ARITHMETIC.  35 

Remark  i.  —  Quantities  such  as  area,  volume,  velocity,  electric 
current,  etc.,  for  which  derived  units  are  used,  may  be  called  de- 
rived quantities  for  the  reason  that  they  are  defined  (as  quantities) 
in  terms  of  the  fundamental  quantities,  length,  mass,  and  time. 
For  example,  the  density  of  a  body  is  defined  as  the  ratio  of  its 
mass  to  its  volume  ;  the  velocity  of  a  body  is  defined  as  the 
quotient  obtained  by  dividing  the  distance  traveled  during  an  in- 
terval of  time,  by  the  interval,  etc. 

Remark  2. — There  is  much  latitude  in  the  choice  of  funda- 
mental units.  A  single  fundamental  unit  would  be  theoretically 
sufficient,  inasmuch  as  it  is  possible  to  define  all  physical  units  in 
terms  of  any  one.  The  choice  of  fundamental  units  is  a  matter 
which  is  governed  solely  by  practical  considerations  ;  in  the  first 
place  the  fundamental  units  must  be  easily  preserved  as  material 
standards,  and  in  the  second  place  the  fundamental  quantities 
must  be  susceptible  of  very  accurate  measurement,  for  the  defini- 
tion of  a  derived  unit  cannot  be  realized'^  with  greater  accuracy 
than  the  fundamental  quantities  can  be  measured. 

13.  The  c.  g.  s.  system  of  units.!  —  Derived  units  based  upon 
the  centimeter  as  the  unit  length,  the  gram  as  the  unit  mass,  and 
the  second  as  the  unit  time,  are  in  common  use.  This  system  of 
derived  units  is  called  the  c.  g.  s.  (centimeter-gram-second) 
system. 

Thus  the  square  centimeter  is  the  c.  g.  s.  unit  of  area,  the 
cubic  centimeter  is  the  c.  g.  s.  unit  of  volume,  one  gram  per 
cubic  centimeter  is  the  c.  g.  s.  unit  of  density,  one  centimeter  per 
second  is  the  c.  g.  s.  unit  of  velocity,  etc. 

*  The  definition  of  a  physical  quantity  is  always  an  actual  physical  operation. 
Thus  the  mass  of  a  body  is  defined  by  the  operation  of  weighing  with  a  balance  ;  the 
density  of  a  body  is  defined  by  the  operations  involved  in  the  finding  of  mass  and 
volume,  for  mass  and  volume  must  be  determined  before  mass  can  be  divided  by 
volume  to  give  density. 

f  So  long  as  the  English  units  of  length  and  mass  continue  to  be  used,  it  will  be 
necessary  for  engineers  to  use  the  units  of  the  f.  p.  s.  (  foot-pound-second)  system  to 
some  extent,  although  these  systematic  f.  p.  s.  units  are,  many  of  them,  never  used  in 
commercial  work.  Thus  the  f.  p.  s.  unit  of  force  is  \h&  poundal,  the  f.  p.  s.  unit  of 
work  is  \ht  foot-poundal. 


36   .  ELEMENTS    OF   MECHANICS. 

Practical  units.  —  In  many  cases  the  c.  g.  s.  unit  of  a  quantity 
is  either  inconveniently  small  or  inconveniently  large  so  that  the 
use  of  the  c.  g.  s.  unit  would  involve  the  use  of  very  awkward 
numbers.  Thus  the  power  required  to  drive  a  small  ventilating 
fan  is  500,000,000  ergs  per  second,  the  electrical  resistance  of 
an  ordinary  incandescent  lamp  is  220,000,000,000  *  c.  g.  s. 
units  of  resistance,  the  capacity  of  an  ordinary  Leyden  jar  is 
0.000,000,000,000,000,005  c.  g.  s.  units  of  capacity.  In  such 
cases  it  is  convenient  to  use  a  multiple  or  a  sub-multiple  of  the 
c.  g.  s.  unit  as  a  practical  unit.  Thus  5x10^  ergs  per  second 
is  equal  to  50  watts,  22  x  lo^*^  c.  g.  s.  units  of  resistance  is  220 
ohms,  5  X  lO"^^  c.  g.  s.  units  of  electrostatic  capacity  is  equal  to 
0.005  micro-farad. 

Legal  units.  —  The  system  of  units  now  in  general  use  presents  several  cases  in 
which  the  fundamental  measurement  of  a  derived  quantity  in  terms  of  length,  mass, 
and  time  is  extremely  laborious  and  not  very  accurate  at  best.  Thus  the  measurement 
of  electrical  resistance  in  terms  of  length,  mass,  and  time  is  very  difficult,  whereas  the 
measurement  of  electrical  resistance  in  terms  of  the  resistance  of  a  given  piece  of  wire 
is  very  easy  indeed  and  it  may  be  carried  out  with  great  accuracy.  In  every  such  case 
the  fundamental  measurement  is  carried  out  once  for  all  with  great  care  and  the  best 
possible  material  copy  is  made  of  the  derived  unit  and  this  copy  is  adopted  as  the 
standard  legal  unit. 

14.  Dimensions  of  derived  units. — The  definition  of  a  derived 
unit  always  implies  an  equation  which  involves  the  derived  unit 
together  with  one  or  more  of  the  fundamental  units  of  length, 
mass,  and  time.  This  equation  solved  for  the  derived  unit  is  said 
to  express  the  dimensions  of  that  unit.f  Thus  the  velocity  of  a 
body  is  defined  as  the  quotient  ///,  where  /  is  the  distance  traveled 
by  the  body  during  the  interval  of  time  /,  so  that  the  unit  of 
velocity  is  equal  to  the  unit  of  length  divided  by  the  unit  of  time. 

Examples.  —  Let  /  be  the  unit  of  length,  in  the  unit  of  mass, 
and  /  the  unit  of  time.  Then  the  unit  of  area  is  equal  to  /^,  the 
unit  of  volume  is  equal  to  /^,  the  unit  of  density  is  equal  to    m\l^ 

*  In  the  writing  of  very  large  or  very  small  numbers  it  is  always  more  convenient 
and  more  intelligible  to  use  a  positive  or  negative  power  of  lo  as  a  factor.  Thus 
220,000,000,000  is  best  written  as  22  X  io'°>  and  0.000,000,000,000,000,005  is  best 
written  as  5  X  io~^^- 

t  And  also  the  dimensions  of  the  derived  quantity. 


PHYSICAL  ARITHMETIC.  17 

the  unit  of  velocity  is  equal  to  //^,  the  unit  of  force  is  equal  to 
mH^y    the  unit  magnetic  pole  is  equal  to    t/m/^//,    etc. 

Naming  of  derived  units.  —  Many  derived  units  have  received 
specific  names.  Such  are  the  dyne,  the  erg,  the  ohm,  the  ampere, 
the  volt,  etc.  Those  derived  units  which  have  not  received  spe- 
cific names  are  specified  by  writing,  or  speaking-out,  their  dimen- 
sions. Thus  the  unit  of  area  is  the  square  centimeter,  the  unit 
of  density  is  the  gram  per  cubic  centimeter,  the  unit  of  velocity  is 
the  centi^neter  per  second,  the  unit  of  momentum  is  the  gram- 
centimeter  per  second  (written  gr.  cm. /sec).  In  the  case  of  units 
which  have  complicated  dimensions  this  method  is  not  conven- 
ient in  speech.  Thus  we  specify  a  certain  magnetic  pole  as 
150  gr.^  cm.  V  sec.  (spoken,  150  c.  g.  s.  units  pole). 

15.  Scalar  and  vector  quantities.  —  A  scalar  quantity  is  a  quan- 
tity which  has  magnitude  only.  Thus  everyone  recognizes  at 
once  that  to  specify  10  cubic  meters  of  sand,  25  kilograms  of 
sugar,  5  hours  of  time,  is,  in  each  case,  to  make  a  complete  spec- 
ification. Volume,  mass,  time,  energy,  electric  charge,  etc.,  are 
scalar  quantities. 

A  vector  quantity  is  a  quantity  which  has  both  magnitude  and 
direction,  and  to  specify  a  vector  one  must  give  both  its  magnitude 
and  direction.  This  necessity  of  specifying  both  the  magnitude 
and  direction  of  a  vector  is  especially  evident  when  one  is  con- 
cerned with  the  relationship  of  two  or  more  vectors.  Thus  if  one 
travels  a  stretch  of  10  kilometers  and  then  a  stretch  of  5  kilo- 
meters more,  he  is  by  no  means  necessarily  i  5  kilometers  from 
home  ;  his  position  is,  in  fact,  indeterminate  until  the  direction  of 
each  stretch  is  specified.  If  one  man  pulls  on  a  car  with  a  force 
A  of  200  units  and  another  pulls  with  a  force  B  oi  \oo  units,  the 
total  force  acting  on  the  car  is  by  no  means  necessarily  equal  to 
300  units.  In  fact,  the  total  force  is  unknown  both  in  magnitude 
and  direction  until  the  direction  as  well  as  the  magnitude  of 
each  force  A  and  B  is  specified.  Length,  velocity,  accelera- 
tion, momentum,  force,  magnetic  field  intensity,  etc.,  are  vector 
quantities. 


38  ELEMENTS   OF   MECHANICS. 

Representation  of  a  vector  by  a  line.  —  In  all  discussions  of 
physical  phenomena  which  involve  the  relationships  of  vectors,  it 
is  a  great  help  to  the  understanding  to  represent  the  vectors  by 
lines.  Thus  in  the  discussion  of  the  combined  action  of  several 
forces  on  a  body,  it  is  a  great  advantage  to  represent  each  force 
by  a  line.  To  represent  a  vector  by  a  line,  draw  the  line  in  the 
direction  of  the  vector  (from  any  convenient  point)  and  make  the 
length  of  the  line  proportional  to  the  magnitude  of  the  vector. 
Thus  if  a  northward  velocity  of  600  centimeters  per  second  of  a 
moving  body  is  to  be  represented  by  a  line,  draw  the  line  to  the 
north  and  let  each  unit  length  of  line  represent  a  chosen  number 
of  units  of  velocity. 

When  a  vector  a  is  represented  by  a  line,  the  line  is  parallel  to 
a  and  the  value  of  a  is  given  by  the  equation 

in  which  /  is  the  length  of  the  line  and  5  is  the  number  of  units 
of  a  represented  by  each  unit  length  of  the  line.  The  quantity 
S  is  called  the  scale  to  which  the  line  represents  the  vector  a. 

16.  Addition  of  vectors.  The  addition  polygon.  —  Many  cases 
arise  in  physics  where  it  is  necessary  to  consider  the  single  force 
which  is  equivalent  to  the  combined  action  of  several  given  forces  ; 
where  it  is  necessary  to  consider  the  single  actual  velocity  which  is 
equivalent  to  several  given  velocities  each  produced,  it  may  be, 
by  a  separate  cause ;  where  it  is  necessary  to  consider  the  single 
acttial  intensity  of  a  magnetic  field  due  to  the  combined  action  of 
several  causes  each  of  which  would  alone  produce  a  magnetic 
field  of  given  direction  and  intensity ;  and  so  on.  The  single 
vector  is  in  each  case  called  the  vector-sum,  or  resultant,  of  the 
several  given  vectors.  Scalar  quantities  are  added  by  the  ordi- 
nary methods  of  arithmetic,  thus  10  pounds  of  sugar  plus  15 
pounds  of  sugar  is  25  pounds  of  sugar;  but  the  addition  of 
several  vectors  is  not  an  arithmetical  operation,  it  is  a  geomet- 
rical operation,  and  it  is  for  this  reason  that  the  addition  of  vectors 
is  sometimes  called  geometric  addition.      In  order  to  make  the 


PHYSICAL   ARITHMETIC. 


39 


following  discussion  easily  intelligible  it  is  made  to  refer  to  the 
special  case  of  the  addition  of  forces. 

Addition  of  two  forces.     The  parallelogram  of  forces.  —  Let  the 
lines  a  and  ^,  Fig.  9,  represent  two  forces  acting  upon  any  body, 


Fig.  10. 

a  boat  for  example.  The  vector-sum  or  resultant  of  the  two 
forces  a  and  b  is  represented  by  the  diagonal  r  of  the  parallelo- 
gram of  which  a  and  b  are  the  sides.  It  is  evident  that  the 
geometric  relation  between  a^  b  and  r  is  completely  represented 
by  the  triangle  in  Fig.  10,  in  which  the  line  which  represents  the 
force  b  is  drawn  from  the  extremity  of  the  line  which  represents 
the  force  a. 

Additio7t  of  any  number  of  forces.  The  force  polygon.  *  —  Given 
a  number  of  forces  a^  b^  c  and  d.  Draw  the  line  which  represents 
the  force  a  from  a  chosen  point 
(7,  Fig.  II,  draw  the  line  which 
represents  the  force  b  from  the 
extremity  of  a^  draw  the  line 
which  represents  c  from  the  ex- 
tremity of  b^  and  draw  the  line 
which  represents  the  force  d  from 
the  extremity  of  c.  Then  the  line 
from  O  to  the  extremity  of  d  rep- 
resents the  geometric  sum  of  the  forces  ^,  ^,  c  and  d. 

The  vector  sum  of  a  number  of  forces  is  equal  to  zero  if  the  forces 
are  parallel  and  proportional  to  the  sides  of  a  closed  polygon,  the 

*  The  sides  of  a  force  polygon  need  not  all  lie  in  a  plane  except  of  course  when 
the  polygon  is  a  triangle.  Three  forces  in  equilibrium  must  not  only  be  parallel  to  a 
certain  plane,  but  their  lines  of  action  must  actually  lie  in  one  plane,  otherwise  the 
forces  will  have  an  unbalanced  torque  action. 


Fig.  11, 


40 


ELEMENTS   OF   MECHANICS. 


directions  of  the  forces  being  in  the  directions  in  which  the  sides  of 
the  polygo7i  would  be  traced  in  going  round  the  polygo7t. 

A  particular  case  of  this  general  proposition  is  that  the  vector 
sum  or  resultant  of  three  forces  is  equal  to  zero  if  the  three  forces 
are  parallel  and  proportional  to  the  three  sides  of  a  triangle  and 
in  the  direction  in  which  the  sides  would  be  passed  over  in  going 
round  the  triangle. 

Note.  —  What  is  said  above  concerning  the  addition  of  forces 
applies  to  the  addition  of  vectors  of  any  kind,  velocities,  accelera- 
tions, magnetic  field  intensities  and  so  on. 

17.  Resolution  of  vectors.  —  Any  vector  may  be  replaced  by  a 
number  of  vectors  of  which  it  is  the  sum.  The  simplest  case  is 
that  in  which  a  vector  is  replaced  by  two  vectors  which  are  par- 
allel and  proportional  to  the  respective  sides  of  a  parallelogram, 
of  which  the  diagonal  represents  the  given  vector.     If  a  rectangle 


■^<r 


be  constructed  whose  diagonal  represents  a  given  vector,  then  the 
sides  of  the  rectangle  will  represent  what  are  called  the  rectangu- 
lar components  of  that  vector  in  the  directions  of  those  sides. 

Example.  —  Consider  a  force  F,  Fig.  1 2,  pulling  on  a  car  which 
is  constrained  to  move  along  a  track.  The  force  F  is  equivalent 
to  a  and  ^  combined.  The  force  /3  has  no  effect  in  moving  the 
car,  so  that  a  is  the  effective  part  of  F.  This  force  a  is  called 
the  resolved  part  of  F,  or  the  component  of  F  in  the  direction  of 
the  track. 

18.  Scalar  and  vector  products  and  quotients.  —  The  product,  or  quotient,  of  a 
scalar  and  a  vector,  a,  is  another  vector  parallel  to  a. 

Examples.  —  The  distance  /  traveled  by  a  body  in  time  /  is  equal  to  the  product 
vt,    v^rhere  v  is  the  velocity  of  the  body  ;  and  /  is  parallel  to  v. 


PHYSICAL   ARITHMETIC.  4 1 

The  force  F  with  which  a  fluid  pushes  on  an  exposed  area  a  is  equal  to  the  prod- 
uct pa  where  p  is  the  hydrostatic  pressure  of  the  fluid  (a  scalar).  The  vector 
direction  of  an  area  is  the  direction  of  its  normal  and  this  is  parallel  to  F. 

A  force  7^  which  does  an  amount  of  work  W'va  moving  a  body  a  distance  d  (which 
is  parallel  \.o  F)  is  equal  to    Wld. 

The  acceleration  of  a  body  is  equal  to  Lv\Lt  where  At;  is  the  increment  of 
velocity  in  the  time  interval    A/  ;    the  acceleration  is  a  vector  which  is  parallel  to   Az/. 

19.  Vector  products.  —  Case  I.  Parallel  vectors.  — The  product  or  quotient  of 
parallel  vectors  is  a  scalar.  Thus  W-=^  Fd,  in  which  W  is  the  work  done  by  a 
force  F  acting  through  a  distance  d  in  its  direction  ;  p  =  Fj a,  in  which  p  is  the 
pressure  in  a  liquid  which  exerts  a  force  F  on  an  exposed  area  a  ;  V=  la,  in  which 
V  is  the  volume  of  a  prism  of  base  a  and  altitude  /. 

Case  II.  Orthogonal  vectors.  —  The  product,  or  quotient,  of  two  mutually  per- 
pendicular vectors  is  a  third  vector  at  right  angles  to  both  factors.  Thus  a=  lb  and 
b^=i  aj I,  in  which  a  is  the  area  of  a  rectangle  of  length  /  and  breadth  b  ;  7^=  Fl, 
in  which  T  is  the  moment  or  torque  of  a  force  Fy  and  /  is  its  arm.  The  product  of 
a  vector  and  a  line  perpendicular  thereto  is  called  a  moment  of  the  vector. 

Case  III.  Oblique  vectors.  —  The  product  of  two  oblique  vectors  consists  of  two 
parts,  one  of  which  is  a  scalar  and  the  other  is  a  vector.     Consider  two  vectors,  a 


Fig.  13. 


and  ^,  Fig.  13.  Resolve  /?  into  two  components,  ^'  and  ^'^  respectively  parallel  to 
and  perpendicular  to  a.     Then 

a/3  =  a(^'  +  /3'0=«.^'  +  «i^^ 

in  which  a/5''  is  a  scalar  and  a,^"  is  a  vector.  The  scalar  part  of  a  vector  product  is 
indicated  thus,  S-  a/5  (read  scalar-alpha-beta).  The  vector  part  of  a  vector  product 
is  indicated  thus,  V-  a/5  (read  vector-alpha-beta').  When  F-a/5  =0,  a  and  ^  are 
parallel ;  when  .S  •  a/5  =  o,  a  and  /5  are  orthogonal. 

Examples  of  products  of  vectors. — The  area  of  any  parallelogram  is  equal  to 
¥•  bl  where  b  and  /  are  the  two  sides  of  the  parallelogram.  The  work  done  by  a 
force  is  equal  to  S  •  Fd  where  F  is  the  force  and  d  is  the  displacement  of  the  point 
of  application  of  the  force.  The  volume  of  any  parallopiped  is  equal  to  S  •  al  where 
a  is  the  area  of  the  base  and  /  is  the  length  of  the  other  edge.  The  torque  action  of 
any  force  is  equal  to    V-  Fr   where  F  is  the  force  and  r  is  the  lever  arm. 

20.  Constant  and  variable  quantities.  The  study  of  those 
physical  phenomena  which  are  associated  with  unvarying  con- 


42  ■  ELEMENTS    OF    MECHANICS. 

ditions  is  comparatively  simple,  whereas  the  study  of  those 
phenomena  which  are  associated  with  rapidly  varying  conditions 
is  generally  very  complicated.  Thus,  to  design  a  bridge  is  to  so 
proportion  its  members  that  a  minimum  amount  of  material  may 
be  required  to  build  it,  and  it  is  a  comparatively  simple  problem 
to  design  a  bridge  to  carry  a  steady  load  because  it  is  easy  to 
calculate  the  stress  in  each  member  due  to  a  steady  load,  but  it 
is  an  extremely  complicated  problem  to  design  a  bridge  to  carry 
a  varying  load,  such  as  a  moving  locomotive  which  comes  upon 
the  bridge  suddenly  and  moves  rapidly  from  point  to  point. 

Two  kinds  of  variations  are  to  be  distinguished,  namely,  varia- 
tions in  space  and  variations  in  time. 

Variations  in  time.  —  In  the  study  of  phenomena  which  de- 
pend upon  conditions  which  vary  in  time,  that  is,  upon  conditions 
which  vary  from  instant  to  instant,  it  is  necessary  to  direct  the 
attention  to  what  is  taking  place  at  this  or  that  instant,  or,  in  other 
words,  to  direct  the  attention  to  what  takes  place  during  very 
short  intervals  of  time,  or,  borrowing  a  phrase  from  the  photog- 
rapher, to  make  snap-shots,  as  it  were,  of  the  varying  conditions. 

Definition  of  rate  of  change.  *  —  Principle  of  continuity.  In 
order  to  establish  the  rather  difficult  idea  of  instantaneous  rate 
of  change  of  a  varying  quantity,  it  is  a  great  help  to  make  use 
of  a  simple  physical  example.  Therefore  let  us  consider  a  pail 
out  of  which  water  is  flowing  through  a  hole  in  the  bottom.     Let 

*  Nearly  everyone  falls  into  the  idea  that  such  an  expression  as  lo  feet  per  second 
means  lo  feet  of  actual  movement  in  an  actual  second  of  time,  but  a  body  moving  at 
a  velocity  of  lo  feet  per  second,  might  not  continue  to  move  for  a  whole  second,  or 
its  velocity  might  change  before  a  whole  second  has  elapsed.  Thus,  a  velocity  of  lo 
feet  per  second  is  the  same  thing  as  a  velocity  of  864,000  feet  per  day,  and  a  body 
need  not  move  steadily  for  a  whole  day  in  order  to  move  at  a  velocity  of  864,000  feet 
per  day.  Neither  does  a  man  need  to  work  for  a  whole  month  to  earn  money  at  the 
rate  of  60  dollars  per  month,  nor  for  a  whole  day  to  earn  money  at  the  rate  of  2  dol- 
lars per  day.  A  falling  body  has  a  velocity  of  19,130,000  miles  per  century  after  it 
has  been  falling  for  one  second,  but  to  specify  its  velocity  in  miles  per  century  does  not 
mean  that  it  moves  as  far  as  a  mile  or  that  it  continues  to  move  for  a  century!  The 
units  of  length  and  time  which  appear  in  the  specification  of  a  velocity  are  completely 
swallowed  up,  as  it  were,  in  the  idea  of  velocity,  and  the  same  thing  is  true  of  the 
specification  of  any  rate. 


PHYSICAL   ARITHMETIC.  43 

X  be  the  amount  of  water  in  the  pail.  Evidently  ;ir  is  a  changing 
quantity.  Let  b^x  be  the  amount  of  water  which  flows  out  of 
the  pail  during  a  given  interval  of  time  A/,  then  the  quotient 
A,r/A/  is  called  the  average  rate  of  change  of  x  during  the  given 
interval  of  time,  and,  if  the  interval  A^  is  very  short,  the  quotient 
Lxj^t  approximates  to  what  is  called  the  rate  of  change  of  x  at 
a  given  instant,  or  the  instantaneous  rate  of  change  of  x.  If  the  ' 
amount  of  water  in  the  pail  were  to  change  by  sudden  jumps, 
as  it  were,  then  the  rate  of  change  of  ;r  at  a  given  instant  would 
be  unthinkable  ;  hwt  physical  quantities  which  vary  in  value  from 
instant  to  instant  always  vary  continuously ,  and  have,  therefore,  at 
each  instant,  a  definite  rate  of  change  ;  that  is  to  say,  the  quotient 
^xj^t  always  approaches  a  definite  limiting  value  as  the  interval 
A/  is  made  shorter  and  shorter.  The  amount  of  water  which 
flows  out  of  a  pail  during  a  short  interval  of  time  is  nearly  pro- 
portional to  the  time ;  and  if  the  time  interval  is  very  short,  the 
amount  of  flow  is  more  and  more  nearly  in  exact  proportion  to 
the  time,  so  that  the  quotient  A;r/A/  approaches  a  perfectly  defi- 
nite finite  value  as  A/  and  ^x  both  approach  zero.  Let  it  be  un- 
derstood that  this  paying  attention  to  what  takes  place  during 
very  short  intervals  of  time  does  not  refer  to  observation  but 
to    thinking,     it    is    a    matter    of   mathematics,*    and    therefore 

"^  Two  distinct  methods  are  involved  in  the  directing  of  the  attention  to  what  takes 
place  during  infinitesimal  time  intervals  or  infinitesimal  regions  of  space. 

(«)  The  j7iethod  of  differential  calculus. — A  phenomenon  may  be  prescribed  as 
a  pure  assumption  and  the  successive  instantaneous  aspects  derived  from  this  prescrip- 
tion. Thus  we  may  prescribe  uniform  motion  of  a  particle  in  a  circular  path,  and 
then  proceed  to  analyze  this  prescribed  motion  as  exemplified  in  Art.  33  ;  or  we  may 
prescribe  a  uniform  twist  in  a  cylindrical  metal  rod,  and  then  proceed  to  analyze  the 
prescribed  distortion.  This  method  is  also  illustrated  by  the  example  given  in  the 
text,  in  which  the  expression  for  the  growing  sides  of  a  square  is  prescribed. 

(^)  The  method  of  integral  calculus. — It  frequently  happens  that  we  know  the 
action  that  takes  place  at  a  given  instant  or  in  a  small  region,  and  can  formulate  this 
action  without  difficulty,  and  then  the  problem  is  to  build  up  an  idea  of  the  result  of 
this  action  throughout  a  finite  interval  of  time,  or  throughout  a  finite  region  of  space. 
For  example,  a  falling  body  gains  velocity  at  a  known  constant  rate  at  each  instant, 
how  much  velocity  does  it  gain  and  how  far  does  it  travel  in  a  given  finite  interval  of 
time? 


44    ■  ELEMENTS   OF   MECHANICS. 

the  following  purely  mathematical  illustration  is  a  legitimate 
example. 

Example.  Consider  a  square  of  which  the  sides  are  growing 
in  proportion  to  elapsed  time  so  that  the  length  of  each  side  of 
the  square  may  be  expressed  as  kt^  where  y^  is  a  constant  and  t 
is  elapsed  time  reckoned  from  the  instant  when  one  of  the  sides  is 
equal  to  zero.  Then  the  area  of  the  square  is  5  =  k^f',  and  it  is 
evident  that  the  area  is  increasing.  Let  /  +  A/  be  written  for  / 
in  the  expression  for  5  and  we  have 

5  +  A5  =  k\t  +  A/)2  =  J^t''  -f  2kH'b.t  -f  k\t^t)'' 

whence,  subtracting  5  =  k^t^^  member  from  member,  we  have 

or 

A/ 

from  which  it  is  evident  that  AS/Al  becomes  more  and  more 
nearly  equal  to  2Pt  as  At  is  made  smaller  and  smaller.  The 
value  of  AS/ At  for  an  indefinitely  small  value  of  At  is  usually 
represented  by  the  symbol  *  dSjdt  so  that  we  have 

Propositions  concerning  rates  of  change.  —  {a)  Consider  a  quan- 
tity X  which  changes  at  a  constant  rate  a^  then  the  total  change 
of  X  during  time  t  is  equal  to  at.  For  example,  a  man  earns 
money  at  the  rate  of  2  dollars  per  day  and  in  lo  days  he  earns 
2  dollars  per  day  multiplied  by  lO  days  which  is  equal  to  20 
dollars.  A  falling  body  gains  32  feet  per  second  of  velocity  every 
second,  that  is,  at  the  constant  rate  of  32  feet  per  second  per 
second,  and  in  3  seconds  it  gains  32  feet  per  second  per  second 
multipHed  by  3  seconds  which  is  equal  to  96  feet  per  second. 

■*The  symbol  dSjdi  is  one  single  algebraic  symbol  and  it  is  not  to  be  treated 
otherwise.  It  stands  for  the  rate  of  change  of  S  at  any  given  instant  Sind  is  to  be  so 
read. 


PHYSICAL   ARITHMETIC.  45 

(b)  Consider  a  quantity  y  which  always  changes  k  times  as  fast 
as  another  quantity  x,  then,  if  the  two  quantities  x  and  y  start 
from  zero  together,  y  will  be  always  k  times  as  large  as  x.  That 
is,  if  dyldt  =  k  •  dxjdt^  then  y  =n  kx  if  y  and  x  start  from  zero 
together. 

Conversely,  if  one  quantity  is  always  k  times  as  large  as  an- 
other it  must  always  change  ^  times  as  fast. 

(c)  Consider  a  quantity  s  which  is  equal  to  the  sum  of  a  num- 
ber of  varying  quantities  x,  y  and  ^,  then  the  rate  of  change  of  s 
is  equal  to  the  sum  of  the  rates  of  change  of  x,  and  j,  and  ^. 
This  may  be  shown  as  follows  :  let  Ax,  Ay  and  A^  be  the  in- 
crements of  X,  y  and  ^  during  a  given  interval  of  time  Af,  then 
the  increment  of  s  is 

As  =  Ax  -i-  Ay  -j-  A^ 

whence,  dividing  both  members  by  At  we  have 

A^       Ax      Ay      A-s" 
a7^  A7"^  A7  '^At 

or,  if  the  interval  A^  is  very  short,  we  have 


ds      dx       dy      dz 
dt       dt^  dt  ^  dt 


(i) 


Conversely,  if  the  relation  (i)  is  given,  that  is,  if  ^  is  a  quantity 
whose  rate  of  change  is  known  to  be  equal  to  the  sum  of  the 
rates  of  change  of  x,  and  y,  and  z,  then  s  must  be  equal  to 
X  -\-  y  +  ^    '^^  ^,  ^y  y,  and  z  all  start  from  zero  together. 

Variations  in  space.  Imagine  a  bar  of  iron  one  end  of  which 
is  red  hot  and  the  other  end  of  which  is  cold.  Evidently  the 
temperature  of  the  bar  varies  from  point  to  point.  The  pressure 
in  a  vessel  of  water  increases  more  and  more  with  the  depth,  the 
density  of  the  air  decreases  more  and  more  with  increasing  alti- 
tude above  the  sea. 

Such  quantities  as  temperature,  pressure,  and  density  which 
refer  to  the  physical  conditions  at  various  points  in  a  substance 


46  ELEMENTS   OF   MECHANICS. 

are  called  distributed  quantities.  The  distribution  is  said  to  be 
uniform  when  the  quantity  has  the  same  value  throughout  a  sub- 
stance, the  distribution  is  said  to  be  non-uniform  when  the  quan- 
tity varies  in  value  from  point  to  point.  Thus  the  temperature 
of  the  air  in  a  room  is  uniform  if  it  is  the  same  throughout, 
whereas  the  temperature  of  a  bar  of  iron  which  is  red  hot  at  one 
end  and  cold  at  the  other  end  is  non-uniform. 

In  the  study  of  phenomena  dependent  upon  conditions  which 
vary  from  point  to  point  in  space,  the  attention  must  be  directed  to 
what  takes  place  in  very  small  regions^  because  too  much  takes 
place  in  a  finite  region.  Let  it  be  understood,  however,  that  this 
paying  attention  to  what  takes  place  in  small  regions  does  not 
refer  to  observation  but  to  thinking,  it  is  a  matter  of  mathematics, 
and  therefore  the  following  illustration  is  a  legitimate  example, 
even  though  the  physics  may  not  be  entirely  clear : 

A  rigid  wheel  rotates  at  a  speed  of  n  revolutions  per  second. 
Let  us  consider  what  is  called  the  kinetic  energy  of  the  wheel. 
Now  the  kinetic  energy  of  a  moving  body  is  equal  to  ^mv^,  where 
m  is  the  mass  of  the  body  in  grams  and  v  is  its  velocity  in  centi- 
meters per  second  ;  but  the  difficulty  here  is  that  the  different 
parts  of  the  wheel  have  different  velocities,  and  if  we  are  to  apply 
the  fundamental  formula  for  kinetic  energy  (=  ^jnv^)  to  a  rotating 
wheel  it  is  necessary  to  consider  each  small  portion  of  the  wheel 
by  itself  Thus,  a  small  portion  of  the  wheel  at  a  distance  r  from 
the  axis  has  a  velocity  which  is  equal  to  27rnr,  and,  if  we  repre- 
sent the  mass  of  the  small  portion  by  Aw,  the  kinetic  energy 
of  the  portion  will  be  J  x  Am  x  {27rnrf,  or  iir^n^  •  r^Am  ;  so 
that  the  total  kinetic  energy  of  the  wheel  will  be  equal  to  the 
sum  of  a  large  number  of  such  terms  as  this.  But,  the  factor 
27r^n^  is  common  to  all  the  terms,  and  therefore  the  total  kinetic 
energy  is  equal  to  iir'^n'^  times  the  sum  of  a  large  number  of  terms 
like  r^Am.     This  sum  is  called  the  moment  of  inertia  of  the  wheel. 

Gradient.  —  Consider  an  iron  bar  of  which  the  temperature  is 
not  uniform.  Let  A  7^  be  the  difference  of  the  temperatures  at 
two  points  distant  Ax  from  each  other,  then  the  quotient  AT  I  Ax 


PHYSICAL   ARITHMETIC.  4/ 

is  called  the  average  temperature  grade  or  gradient  along  the 
stretch  A;tr,  and  if  A;r  is  very  small  the  quotient  AT/ Ax  is  called 
the  actual  temperature  gradient  and  it  is  represented  by  the 
symbol  dTjdx.  The  use  of  this  idea  of  temperature  gradient  is 
illustrated  in  the  discussion  of  the  conduction  of  heat. 

21.  Varying  vectors.*  —  The  foregoing  article  refers  solely  to 
varying  scalar  quantities.  The  mathematics  of  varying  vectors 
may  also  be  considered  in  two  parts,  namely  time  variation  and 
space  variation. 

Time  variation  of  velocity,  f  —  The  velocity  of  a  body  is  defined 
as  the  distance  traveled  in  a  given  time  divided  by  the  time. 
When  a  velocity  always  takes  place  in  a  fixed  direction  it  may  be 
thought  of  as  a  purely  scalar  quantity.  Thus  a  falling  body  has 
a  velocity  of  50  feet  per  second  at  a  given  instant  and  3  seconds 
later  it  has  a  velocity  of  146  feet  per  second,  so  that  the  increase 
of  velocity  in  3  seconds  is  96  feet  per  second  and  the  rate  of  in- 
crease is  96  teet  per  second  divided  by  3  seconds  which  is  equal 
to  32  feet  per  second  per  second  ;  but,  suppose  that  the  velocity 
of  a  body  at  a  given  instant  is  50  units  in  a  specified  direction 
and  that  3  seconds  later  it  is  146  units  in  some  other  specified 
direction,  then  the  change  of  velocity  is  by  no  means  equal  to  96 
units  and  the  rate  of  change  of  the  velocity  is  by  no  means  equal 
to  32  units  per  second. 

The  rate  of  change  of  the  velocity  of  a  body  is  called  the 
acceleration  of  the  body.  Consider  any  moving  body,  a  ball 
tossed  through  the  air  for  example,  let  its  velocity,  v^,  at  a  given 

*This  branch  of  mathematics  is  completely  ignored  in  present  undergraduate 
courses,  and  yet  no  one  can  have  a  clear  insight  into  the  phenomena  of  motion  with- 
out having  an  idea  of  the  time  variation  of  velocity,  and  no  one  can  have  a  clear 
insight  into  the  phenomena  of  fluid  motion  and  of  electricity  and  magnetism  without 
having  some  understanding  of  the  space  variation  of  such  vectors  as  fluid  velocity, 
magnetic  field,  and  electric  field. 

t  What  is  here  stated  concerning  the  time  variation  of  velocity  applies  to  the  time 
variation  of  any  vector  whatever.  Thus  if  any  varying  vector  is  represented  to  scale 
by  a  line  drawn  from  a  fixed  point,  then  the  velocity  of  the  end  of  the  line  represents 
the  rate  of  change  of  the  vector  to  the  same  scale  that  the  line  itself  represents  the 
vector. 


48 


ELEMENTS   OF   MECHANICS. 


instant  be  represented  by  the  line  OA,  Fig.  14,  let  its  velocity 
v^  at  a  later  instant  be  represented  by  the  line  OB,  and  let  the 
elapsed  time  interval  be  At.  Now  the  velocity  which  must  be 
added  (geometrically)  to  v^  to  give  v^  is  the  velocity  Av  which  is 
represented  by  the  line  AB.  Therefore,  the  change  of  the  velocity 
of  the  tossed  ball  during  the  interval  At  is  the  vertical  velocity 
Av  shown  in  the  figure,  and  the  acceleration,  a,  of  the  ball  is 
equal  to  AvjAt  which  is  of  course  in  the  direction  of  Av.  If 
the  varying  velocity  of  a  tossed  ball  be  represented  by  a  line  OP, 
Fig.  1 5,  drawn  from  a  fixed  point  0 ;  then,  as  the  velocity  changes, 


Fig.  14. 


Fig.  15. 


the  line  OP  will  change,  the  point  P  will  move,  and  the  velocity 
of  the  point  P  will  represent  the  acceleration  of  the  body  to  the 
same  scale  that  the  line   OP  represents  the  velocity  of  the  body. 

The  orbit  of  a  moving  body  is  the  path  which  the  body  de- 
scribes in  its  motion.  Thus  the  orbit  of  a  tossed  ball  is  a  para- 
bola as  shown  in  Fig.  \6a,  and  the  orbit  of  a  ball  which  is  twirled 
on  a  cord  is  a  circle  as  shown  in  Fig.  ija.  Suppose  a  line  OP, 
Fig.  16^  or  Fig.  ijb,  drawn  from  a  fixed  point,  be  imagined  to 
change  in  such  a  way  as  to  represent  at  each  instant  the  velocity 
of  the  body  as  it  describes  its  orbit,  then  the  end  P  of  the  line 
will  describe  a  curve  called  the  hodograph  of  the  orbit  and,  of 
course,  the  velocity  of  the  point  P  will  represent  at  each  instant 
the  acceleration  of  the  moving  body.  Thus  the  hodograph  of  a 
tossed  ball  is  a  vertical  straight  line  as  shown  in  Fig.  \6b,  this  is 
evident  when  we  consider  that  a  tossed  ball  has  a  constant  ac- 
celeration vertically  downwards,  so  that  the  point  P,  Fig.    \6b. 


PHYSICAL   ARITHMETIC. 


49 


must  move  at  a  constant  velocity  vertically  downwards.  The 
hodograph  of  a  ball  twirled  on  a  string  is  a  circle  as  shown  in 
Fig.  \jb^  this  is  evident  when  we  consider  that  the  magnitude  of 


Fig.  16«. 


Fig.  163. 


the  velocity  of  the  body  in  Fig.  i  J  a  is  constant,  so  that  the  length 
of  OPy   Fig.  1 7^,  which  represents  the  velocity  of  the  body,  must 


Fig.  17«. 


N. 


\ 


Ofc- 


\ 

\ 

>1P 


,.\- 


'^'*»>,.^     hodograph       x 


Fig.  1 73. 


also  be  constant.     Furthermore,  the  velocity  of  the  ball  is  always 
at  right  angles  to  the  cord  in  Fig.  lya,  and  therefore  the  line 
OPy   Fig.  I  yd,  is  always  at  right  angles  to  the  cord,  so  that  the 
4 


50 


ELEMENTS   OF   MECHANICS. 


generating  point  P  of  the  hodograph  makes  the  same  number  of 
revolutions  per  second  as  the  twirled  ball.  It  is  important  to 
note  also  that  the  velocity  a  of  the  point  P  in  Fig.  i  jb,  is  always 
parallel  to  the  cord  in  Fig.  i  ja,  that  is,  the  acceleration  of  the 
ball  in  Fig.  ija  is  at  each  instant  in  the  direction  of  the  cord. 

Space  variation  of  vectors ,  —  The  simplest  idea  connected  with  the  space  variation 
of  a  vector  is  the  idea  of  the  stream  line,  if  it  may  be  permitted  to  use  the  terminology 
of  fluid  motion  to  designate  a  general  idea.  A  stream  line  in  a  moving  fluid  is  a  line 
drawn  through  the  fluid  so  as  to  be  at  each  point  parallel  to  the  direction  in  which 
the  fluid  is  moving  at  that  point.  Thus  if  a  pail  of  water  be  rotated  about  a  vertical 
axis,  the  stream  lines  are  a  system  of  concentric  circles  as  shown  in  Fig.  i8,  and  Fig. 
19  represents  the  approximate  trend  of  the  stream  lines  in  the  case  of  a  jet  of  water 


Fig.  1 


Fig.  19. 


issuing  from  a  tank.     Where  the  stream  lines  come  close  together  the  velocity  of  the 
water  is  great  and  where  they  are  far  apart  the  velocity  is  small. 

The  potential  of  a  distributed  vector.  —  In  some  simple  cases  of  fluid  motion  it  is 
geometrically  possible  to  look  upon  the  stream  lines  as  the  lines  of  slope  of  an 
imagined  hill  the  steepness  of  which  represents  the  velocity  of  the  fluid  at  each  point 
both  in  magnitude  and  in  direction.  The  height  of  this  imagined  hill  at  a  point  is 
called  the  potential  of  the  fluid  velocity  at  that  point.  The  idea  of  potential  is  espe- 
cially useful  in  the  study  of  electricity  and  magnetism. 


Problems. 

8.  A  table  top  is   10  feet  long  and  50  inches  wide.     Find  its 
area  in  inch-feet,  and  explain  the  result. 

9.  A  body  has  a  mass  of  60  pounds  and  a  volume  of  2  gallons. 
Find  its  density  without  reducing  data  in  any  way. 


PHYSICAL  ARITHMETIC. 


51 


10.  A  water  storage  basin  has  an  area  of  2,000  acres,  find  the 
volume  of  water  in  acre-feet  required  to  fill  the  basin  to  a  depth 
of  16  feet.  Explain  the  acre-foot  as  a  unit  of  volume  and  find 
the  number  of  gallons  in  one  acre -foot. 

11.  A  man  travels  at  a  velocity  of  6  feet  per  second ;  how  far  does 
he  travel  in  two  hours  ?  Find  the  result  without  reducing  the  data 
in  any  way.      Explain  XhQ  foot-hoiir  per  second  as  a  unit  of  length. 

1*2.  A  man  starts  from  a  given  point  and  walks  three  miles  due 
north,  then  two  miles  northeast,  then  two  miles  south,  and  then 
one  mile  east.  Show  by  means  of  a  vector  diagram  how  far, 
and  in  what  direction  he  is  from  his  starting  point. 

13.  A  stream  flows  due  south  at  a  velocity  of  two  miles  per 
hour.  A  man  rows  a  boat  in  an  eastward  direction  at  a  velocity 
of  four  miles  per  hour.  What  is  the  actual  velocity  of  the  boat 
and  in  what  direction  is  it  moving  ? 

14.  A  stream  flows  due  south  at  a  velocity  of  two  miles  per 
hour.  A  man,  who  can  row  a  boat  at  a  velocity  of  four  miles  per 
hour,  wishes  to  reach  the  opposite  bank  at  a  point  due  east  of  his 
starting  point.  Show  by  means  of  a  vector  diagram  the  direction 
in  which  he  must  row. 


L^7 

'8 


[k1   R    Fl    liil  W 


mm 


ir¥ 


WW 


7 

50  feet  pejf  second 

Fig.  16A 


§^MMM 


WWWWE 


15.  An  anemometer  on  board  ship  indicates  a  wind  velocity  of 
28  miles  per  hour  apparently  from  the  northeast.  The  ship,  how- 
ever, is  moving  due  north  at  a  velocity  of  1 5  miles  per  hour. 
What  is  actual  direction  and  velocity  of  the  wind  ? 

16.  A  gun  which  produces  a  projectile- velocity  of  200  feet  per 
second  is  mounted  aboard  a  car  with  its  barrel  G  at  right  angles 
to  the  direction  of  motion  of  the  car,  as  shown  in  Fig.  i6p.     The 


52  ELEMENTS   OF   MECHANICS. 

car  is  traveling  50  feet  per  second.  The  sights  are  to  be  arranged 
at  an  angle  6  to  the  gun  barrel  as  shown.  Find  the  value  of  0 
so  that  the  ball  may  hit  any  object  which,  at  the  instant  of  firing, 
is  in  the  line  S  of  the  sights. 

A^o^e.  —  This  problem  illustrates  the  vector  addition  of  velocities.  The  sight  line 
must  of  course  be  in  the  actual  direction  in  which  the  bullet  is  moving  vv^hen  it  leaves 
the  gun. 

17.  A  body  moves  at  a  velocity  of  20  miles  per  hour  in  a  direc- 
tion 20°  north  of  east';  find  the  northward  and  eastward  com- 
ponents of  its  velocity. 

18.  Find  the  magnitude  and  direction  of  the  single  force  which 
is  equivalent  to  the  combined  action  of  three  forces  A,  B  and  C\ 
force  A  being  northwards  and  equal  to  200  units,  force  B  being 
towards  the  north-east  and  equal  to  1 50  units,  and  force  C  being 
eastwards  and  equal  to  100  units. 

19.  A  horse  pulls  on  a  canal  boat  with  a  force  of  600  pounds- 
weight  and  the  rope  makes  an  angle  of  25°  with  the  line  of  the 
boat's  keel.     Find  the  component  of  the  force  parallel  to  the  keel. 

20.  If  500  grams  of  water  leak  out  of  a  pail  in  26  seconds, 
what  is  the  average  rate  of  leak  ? 

21.  A  man  earns  ^27.50  in  8^  days.  What  is  the  average 
rate  at  which  he  earns  money  ? 

22.  During  28  seconds  the  velocity  of  a  train  increases  from 
zero  to  1 2  feet  per  second.  What  is  the  average  rate  of  increase 
of  velocity? 

23.  A  train  gains  a  speed  of  32  miles  per  hour  in  80  seconds. 
Find  its  average  acceleration  in  miles  per  hour  per  second. 

24.  A  pole  22  feet  long  is  dragged  sidewise  over  a  field  at  a 
velocity  of  8  feet  per  second.  At  what  rate  does  the  pole  sweep 
over  area  ? 

25.  A  prism  has  a  base  of  25  square  cm.,  its  height  is  increas- 
ing at  the  rate  of  5  cm.  per  second.  How  fast  is  its  volume  in- 
creasing ? 

26.  The  slope  of  a  hill  falls  60  feet  in  a  horizontal  distance  of 
270  feet.     What  is  the  grade  ? 


PHYSICAL   ARITHMETIC.  53 

27.  One  side  of  a  brick  wall  is  at  a  temperature  of  o°  C.  and 
and  other  side  is  at  a  temperature  of  23°  C.  The  wall  is  30.5 
cm,  thick.  What  is  the  average  temperature  gradient  through 
the  wall  ? 

28.  At  a  given  point  in  a  water  pipe  the  water  pressure  is  1 1  o 
lbs.  per  square  inch.  Twenty-two  feet  from  this  point  the  pres- 
sure is  75  pounds  per  square  inch.  What  is  the  average  pressure 
gradient  along  the  pipe  ? 


CHAPTER   IV. 
SIMPLE   STATICS. 

22.  Balanced  force  actions.  —  When  a  body  remains  at  rest,  or 
continues  to  move  with  uniform  velocity  along  a  straight  path,  or 
when  a  body  continues  to  rotate  at  uniform  speed  about  a  fixed 
axis,  the  forces  which  act  on  the  body  are  balanced.  The  study 
of  balanced  force  actions,  or  as  it  is  sometimes  expressed,  the  study 
of  forces  in  equilibrium,  constitutes  the  science  of  statics-.  The 
science  of  statics  really  includes  the  study  of  equilibrium  in  its 
widest  sense,  namely,  the  equilibrium  of  the  forces  which  act 
upon  the  parts  of  a  distorted  body  (the  statics  of  elasticity),  and 
the  equilibrium  of  the  forces  which  act  upon  the  parts  of  a  fluid 
(hydrostatics) ;  but  these  branches  of  statics  are  treated  in  subse- 
quent chapters,  and  the  present  chapter  deals  only  with  the  rela- 
tionship between  forces  which  do  not  tend  to  produce  translatory 
motion  or  rotatory  motion.* 

Every  one  knows  that  even  a  single  force  acting  on  a  body 
may  cause  both  translation  and  rotation.  Thus  a  boat  which  is 
pushed  away  from  a  landing  generally  turns  more  or  less  as  it 
moves  away ;  in  this  case,  however,  it  may  be  the  force  action 
of  the  water  on  the  boat  that  causes  the  turning,  but  a  sidewise 
push  on  the  bow  of  the  boat  certainly  produces  both  translation 
and  rotation  irrespective  of  the  force  action  of  the  water. 

From  the  fact  that  even  a  single  force  can  produce  both  trans- 
lation and  rotation,  it  may  seem  as  though  it  would  be  impossible 
to  consider  separately  the  two  effects  of  a  force,  namely,  {a)  ten- 
dency to  produce  translatory  motion  and  {p)  tendency  to  produce 
rotation  ;  but  every  one  knows  that  forces  may  produce  transla- 
tion without  producing  rotation,  and  that  forces  may  produce  ro- 
tation without  producing  translation.  Thus  a  table  may  be 
moved  without  turning,  and  a  top  may  be  set  spinning  without 

*  See  Art.  28. 

54 


SIMPLE   STATICS.  55 

any  perceptible  sidewise  motion.  The  fact  is,  the  two  ten- 
dencies a  and  b  must  be  considered  separately. 

23.  Tendency  to  produce  translatory  motion.  First  condition 
of  equilibrium.  —  In  order  that  a  number  of  forces  may  have  no 
tendency  to  produce  translatory  motion,  it  is  necessary  and  suf- 
ficient that  the  vector  sum  of  the  forces  be  equal  to  zero,  that  is, 
the  forces  must  be  parallel  and  proportional  to  the  sides  of  a 
closed  polygon  and  in  the  directions  in  which  the  sides  of  the 
polygon  would  be  passed  over  in  going  round  the  polygon,  as 
explained  in  Art.  i6.  .  This  statement  of  the  first  condition  of 
equilibrium  leads  directly  to  the  graphical  method  of  solving  a 
problem  in  statics.  When  the  algebraic  method  of  solution  is  to 
be  used,  the  following  statement  of  the  first  condition  of  equilib- 
rium is  preferable. 

Let  each  force  be  resolved  into  rectangular  components  par- 
allel to  chosen  axes  of  reference.  Let  X^,  X^,  JQ,  etc.,  be  the 
;tr-components  of  the  various  forces,  and  let  1^,  Y^,  Y^  etc.,  be 
their  j-components.  Then  if  the  tendency  of  the  forces  to  pro- 
duce translatory  motion  is  zero,  we  have 

X^  +  X^-{-  X^  +  etc.  =  o  1 
and  (i) 

y.+  y^-^y.^  etc.  =  o  j 

It  is  necessary  in  forming  these  equations  to  consider  ^r-compo- 
nents  as  positive  when  they  are  towards  the  right  and  negative 
when  they  are  towards  the  left ;  and  to  consider  j-components 
as  positive  when  they  are  upwards  and  as  negative  when  they  are 
downwards. 

24.  Tendency  to  produce  rotation.  Definition  of  torque.  —  Con- 
sider a  lever  AB,  Fig.  20,  supported  at  the  point  0  and  in 
equilibrium  under  the  action  of  two  forces  F  and  F'  acting  as 
shown,  then  Fa  is  equal  to  F'  a' .  The  product  Fa  nieasures 
the  tendency  of  the  force  F  to  turn  the  lever  in  one  direction 
about  0,  and  the  product  F^a'  measures  the  tendency  of  the 
force  F'  to  turn  the  lever  in  the  other  direction  about  the  point 


-  '    <K 


56 


ELEMENTS   OF   MECHANICS. 


Oy  and,  rightly  considered,  one  of  these  products  should  be  con- 
sidered as  positive  and  the  other  as  negative  so  that  we  may  write 

Fa  +  F'a'  =  o 

The  product  of  a  force  and  the  perpendicular  distance  from  a 
given  point  to  the  line  of  action  of  the  force ^  measures  the  tendency 
of  the  force  to  turn  a  body  about  the  given  point,  and  this  product 
is  called  the  moment  of  the  force  about  the  point,  or  its  torque. 


Fig.  20. 

Second  condition  of  equilibrium.  —  In  order  that  a  number  of 
forces  may  have  no  tendency  to  turn  a  body  it  is  necessary  *  that 
the  sum  of  the  torque  actions  of  the  forces  about  any  chosen  point 
be  equal  to  zero,  torques  tending  to  turn  the  body  in  one  direc- 
tion being  considered  as  positive  and  torques  tending  to  turn  the 
body  in  the  opposite  direction  being  considered  as  negative.  The 
chosen  point  is  called  the  origin  of  moments. 

It  is  often  convenient  to  express  this  second  condition  of  equi- 
librium in  terms  of  the  components  of  the  respective  forces  and 
the  coordinates  of  their  points  of  application.  Consider  a  force 
F  which  acts  on  a  body,  Fig.  2 1 ,  at  the  point  /  of-  which  the 
coordinates  are  x  and  y  as  shown.  Let  X  and  Y  be  the  com- 
ponents of  F,  then  Xy  is  the  torque  action  of  the  component 
X  about  the  origin  0,  and  Yx  is  the  torque  action  of  the  com- 
ponent Y  about  0  ;  and  inasmuch  as  these  torque  actions  are  in 
opposite  directions  they  must  be  considered  as  opposite  in  sign  so 

*This  condition  is  also  a  sufficient  condition  if  the  first  condition  of  equilibrium  is 
satisfied. 


SIMPLE   STATICS. 


57 


that  the  total  torque  action  of  the  force  F  about  0  is  equal  to 
Xy  —  Yx.     The  torque  action  of  each  force  acting  on  a  body 


y-axi&                           J 

^~-^l 

^^ 

1 

^^ 

^^^  "~  — 

^^^\ 

1 

yO""^ 

—  — ■  X.—      —     - 

X 

1 
^1 

y^— 

— _     0\ 

1 - 

g 

/ 

X-axis 

I             — 

■ 

1 

.  ■ 

1 

I          — 

\ 

^- 

\             ~ 

O    — / 

\ 

\ 

_^^ 

^^ 

^ 

Fie.  21. 


may  be  calculated  in  this  way  and  then  the  sum  of  these  torque 
actions  must  be  equal  to  zero.     That  is 


2(^  -  Kf)  =  o 


(2) 


-£B 


Fig.  22. 

Pure  torque.  —  A  number  of  forces  which  act  on  a  body  may 
have  no  tendency  to  produce  translatory  motion  and  still  have  a 
tendency  to  produce  rotation,  or,  in  other  words,  a  number  of 
forces  may  satisfy  the  first  condition  of  equilibrium  and  not  satisfy 
the  second  condition.  Such  a  combination  of  forces  constitutes 
a  pure  torque  and  the  total  torque  action  is  the  same  about  any 
point  whatever.  For  example,  the  two  equal  and  opposite  forces 
which  are  exerted  on  the  handle  of  an  auger,  as  shown  in  Fig. 
22,  constitute  a  pure  torque.  Such  a  pair  of  forces  is  sometimes 
called  a  couple. 


58 


ELEMENTS   OF   MECHANICS. 


25.  Three  forces  in  equilibrium  intersect  at  a  point.  —  The  forces 
must  lie  in  one  plane  in  order  to  satisfy  the  first  condition  of 
equilibrium.  Choose  the  origin  of  moments  at  the  intersection 
of  two  of  the  forces.  Then  the  third  force  must  pass  through 
this  point,  otherwise  it  will  have  an  unbalanced  torque  action 
about  this  point. 

Any  number  of  forces,  not  hi  equilibriti7n,  acting  on  a  body  are 
together  equivalent  to  a  single  force,  which  is  called  their  resultant; 
except  when  the  forces  constitute  a  pure  torque. 

Proof,  —  Given  a  number  of  forces  in  equilibrium.  If  one  of 
these  forces  is  omitted,  the  combined  action  of  the  others  must  be 
equivalent  to  an  equal  and  opposite  force  having  the  same  line  of 
action.  The  exception  is  also  evident,  since  by  omitting  one  of  a 
set  of  forces  in  equilibrium  the  others  cannot  constitute  a  pure 
torque.  The  magnitude  and  direction  of  the  resultant  of  a  num- 
ber of  forces  is  determined  as  their  vector  sum.  The  point  of 
application  of  this  resultant  is  the  point  about  which  the  given 
forces  have  no  torque  action. 

The  center  of  mass  of  a  body  is  the  point  of  application  of  the  resultant  of  the 
parallel  forces  with  which  gravity  acts  upon  the  particles  of  the  body. 

Proof,  —  Let  the  origin  of  coordinates  be  chosen  at  the  center  of  mass  of  the  body 


•  —    -^ 

,^ZE^si  sT--  :^ 

\m=  ^ 

X-axis 

■• 

.          — 

o      —      — 

-^    -= 

—                                                   J 

y-axis 

mg 

> 

Fig.  23. 


(see  Art.  50)  and  let  the  jj/-axis  be  downwards  as  shown  in  Fig.  23.  Consider  a 
particle  of  mass  m  which  is  pulled  downwards  by  the  force  mg  (see  Art.  33).  The 
torque  action  of  this  force  about  the  origin  is  mgx,  and  the  torque  action  of  all  such 
forces  is   ^mgx  or  g^mx.     But   ^mx  is  equal  to  zero  inasmuch  as  the  origin  is  sup- 


SIMPLE   STATICS. 


59 


posed  to  be  at  the  center  of  mass  of  the  body.  Therefore  the  torque  action  of  all  of 
the  forces  m^  sihont  O  is  zero,  and  consequently  O  is  the  point  of  application  of  the 
total  force  with  which  gravity  acts  on  the  body.  The  center  of  mass  of  a  body  is  for 
this  reason  sometimes  called  the. center  of  gravity  of  the  body. 

26.  D'Alembert's  principle.  — The  ease  with  which  the  relation  between  a  num- 
ber of  forces  in  equilibrium  can  be  shown,  especially  by  the  use  of  graphical  methods, 
makes  it  desirable  to  extend  the  idea  of  balanced  forces  to  the  subject  of  dynamics. 
The  principle  on  which  this  can  be  done  was  first  enunciated  by  D'  Alembert  and  it  is 
called  D'Alembert's  principle.  The  following  is  a  statement  of  D'Alembert's  prin- 
ciple as  applied  to  a  particular  case.  A  part  of  a  mechanism  moves  in  a  prescribed 
way  under  the  combined  action  of  a  number  of  given  forces  ;  if  there  be  introduced 
into  the  system ^c^i^ious  forces  which  are  equal  and  opposite  to  the  forces  required  to 
produce  the  known  accelerations,  then  the  system  of  given  forces  together  with  the 
fictitious  forces  will  be  in  equilibrium.  Thus  if  forces  be  imagined  to  act  outwards  on 
every  part  of  a  rotating  metal  hoop  or  ring,  then  these  forces  may  be  thought  of  as 
producing  the  tension  in  the  rotating  hoop,  the  hoop  may  be  thought  of  as  stationary, 
and  the  problem  becomes  a  problem  in  statics.     See  Art.  60. 


Problems.* 

29.  Three  cylinders,  each  12  inches  in  diameter  and  each 
weighing  200  pounds,  lie  in  a  rectangular  trough  of  which  the 
width  is  27  inches.     Assuming  that  the 

force  action  at  each  point  of  contact  is  at 
right  angles  to  the  surface  of  contact, 
find  the  force  action  at  each  point  of 
contact  of  the  cylinders  with  each  other 
and  with  the  trough. 

30.  The  steam  in  an  engine  cylinder 
pushes  on  the  piston  with  a  force  of 
12,000  pounds-weight.  The  positions  and  lengths  of  connecting 
rod  and  crank  are  shown  in  Fig.  30/.  Find  the  force  with  which 
the  cross-head  pushes  sidewise  against  the  guide,  the  thrust  of 
the  connecting  rod,  and  the  torque  in  pound-feet  exerted  on  the 
crank-shaft,  neglecting  friction  throughout. 

■^Several  problems  in  statics,  illustrating  D'Alembert's  principle  and  the  principle 
of  virtual  work,  are  given  at  the  end  of  Chapter  VI.  Unquestionably  it  is  more  in- 
structive to  solve  problems  in  statics  by  the  graphical  method  than  by  the  analytical 
method,  and  no  case  exists  in  practice  for  which  the  graphical  method  is  not  sufficiently 
accurate.  Most  of  the  problems  in  this  chapter  should  be  solved  by  the  graphical 
method,  and,  if  desirable,  the  analytical  solution  may  then  be  required  as  an  additional 


Fig.  29A 


exercise. 


6o 


ELEMENTS   OF   MECHANICS. 


31.  A  rope  15  feet  long  supports  a  1,000-pound  weight  from 
two  supports    SS,  as  shown  in  Fig.   31/.     Find  the  tension  in 


l^'>'>v>w\s'<^^^^w^u'TOw>w^v^^^^\vy,ww 


Fig.  GO/. 


each  part  of  the  rope  and  the  vertical  force  on  each  support, 
neglecting  weight  of  rope. 


—  14  feet 


TTprrrr 


tpoo  pottjtda 

Fig.  3 1>. 

32.  A  right-angled  lever  of  which  the  position  and  dimensions 
are  shown  in  Fig.  32/  carries  a  weight  of  200  pounds  at  the  end 
of  the  24-inch  arm.  Find 
the  value  and  direction  of 
the  force  exerted  on  the 
lever  by  the  point  O,  and 
find  the  value  of  the  force  K 


Note.  — Three  forces  in  equili- 
brium intersect  in  a  point  so  that 
the  line  of  action  of  the  unknown 
force  at  O  passes  through  the  point 
where  the  lines  W  and  F  intersect. 


200  pounds} 


Fig.  32/. 


33.  Find  the  force  F  required  to  draw  a  4,000-pound  wagon 
up  an  inclined  plane  of  the  dimensions  shown  in  Fig.  33/,  neglect- 
ing friction. 


SIMPLE   STATICS. 


6i 


34.  Find  the  force  F,  Fig.  34/,  required  to  draw  a  200-pound 
block  up  an  inclined  plane  of  the  dimensions  shown,  the  coeffi- 
cient of  friction  between  block  and  plane  being  0.2. 

JVole.  —  See  Art.  52  for  an  explanation  of  the  coefficient  of  friction.  The  direction 
of  the  force  which  the  plane  exerts  on  the  moving  block  is  FO,  as  shown  in  Fig.  34/. 

35.  A  wedge  of  which  the  shape  is  indicated  in  Fig.  35/,  is 
pushed  between  two  blocks  A  and  B  with  a  force  F  of  5,000 
pounds-weight.      The  coefficient  of  friction  between  the  wedge 


6  feet 


and  the  blocks  is  0.2.     Find  the  components  at  right  angles  to 
F  of  the  forces  with  which  the  wedge  pushes  on  A  and  B. 

36.  A  block  weighing  100  pounds  rests  on  a  smooth  floor,  the 


Fig.  35/. 


Fig.  36/. 


coefficient  of  friction  between  the  block  and  the  floor  is  0.2,  and 
the  block  is  pushed  along  by  a  stick  S,  as  shown  in  Fig.  36/). 
Find  the  thrust  of  the  stick  in  pounds-weight. 


62 


ELEMENTS   OF   MECHANICS. 


37.  A  ladder  i6  feet  long  and  weighing  loo  pounds  has  its 
center  of  gravity  7  feet  from  its  lower  end  which  rests  on  a  floor 
at  a  distance  of  4  feet  from  a  vertical  wall  against  which  the  lad- 
der rests,  as  shown  in  Fig.  37/.  Assuming  the  force  a  with 
which  the  wall  pushes  on  the  ladder  to  be  horizontal,  find  the 


J 

d  Ig 

1  1 

^1 

o>  K 

I   a 

Ml 

1 

i 

^ 

/^ 

-K 

li 

f 

4M^ 

waU 


Moot 


Fig.  37A 


Fig.  38A 


magnitude  of  a  and  the  direction  and  magnitude  of  the  force  with 
which  the  ladder  pushes  against  the  floor. 

38.  A  forty-foot  beam  arranged  as  shown  in  Fig.  38/  supports 


Fig.  39/. 

a  weight  of  2,000  pounds.     Find  the  pull  of  the  rope  and  the 
thrust  of  the  beam. 

39.  A  stick  aby  Fig.  39/,  lies  across  a  right-angled  trough 
ABC  as  shown.  Find  the  line  in  the  figure  which  must  be  ver- 
tical in  order  that  the  stick  may  be  in  equilibrium  on  the  assump- 
tion that  there  is  no  friction  at  a  and  b. 


SIMPLE  STATICS. 


63 


Note.  —  The  force  of  gravity  on  the  stick  is  a  vertical  force  and  its  point  of  appli- 
cation is  at  the  center  of  the  stick.     Three  forces  in  equilibrium  intersect  in  a  point. 

40.  Suppose  that  the  coefficient  of  friction  at  a  and  b^  Fig.  39/, 
is  o.  I,  find  the  limiting  positions  of  the  vertical  between  which  the 
stick  will  be  in  equilibrium. 

41.  A  uniform  stick  6  feet  long  which  weighs  10  pounds  has  a 
15-pound  weight  hung  i  foot  from  one  end,  a  20-pound  weight 
hung  2  feet  from  the  same  end,  and  a  25-pound  weight  hung  6 
inches  from  the  other  end.  Find  the  point  at  which  the  stick 
can  be  supported  in  a  horizontal  position. 

Note.  —  To  solve  this  problem,  choose  one  end  of  the  stick  as  the  origin  of  moments 
and  let  x  be  the  distance  from  this  end  to  the  point  of  application  of  the  resultant  of 
all  of  the  forces.  Then  the  sum  of  the  moments  of  the  several  forces  about  the  origin 
is  equal  to  the  sum  of  the  forces  multiplied  by  x^  from  which  relation  x  can  be  cal- 
culated. 

42.  A  simple  bridge  truss  consists  of  two  struts  and  a  tie  rod  as 
shown  in  Fig.  42/.     A  weight  ^of  2,000  pounds  hangs  from 


Fig.  42/. 


Fig.  43/. 


the  point  P.  Find  the  compression  in  each  strut,  the  vertical 
pressure  on  each  abutment,  and  the  tension  in  the  tie  rod,  neglect- 
ing weight  of  the  parts  of  the  truss. 

Note.  — To  solve  this  problem,  consider  first  the  equilibrium  of  the  three  forces  at 
Py  two  of  which  forces  are  parallel  to  the  respective  struts  ;  then  consider  the  equi- 
librium of  the  three  forces  at  Q  ;  and  then  consider  the  equilibrium  of  the  three  forces 
at  R.     Assume  the  abutments  to  exert  vertical  forces  on  the  truss. 

43.  A  table  drawer,  36  inches  in  breadth  and  18  inches  in 
depth  (front  to  back),  is  pulled  by  a  force  F  applied  at  a  distance 
X  from  one  corner  as  shown  in  Fig.  43/.  The  drawer  binds  at 
the  two  comers  /  and  q,  and  it  is  required  to  find  the  smallest 


64 


ELEMENTS   OF   MECHANICS. 


value  of  X  for  which  the  drawer  can  be  pulled  out  by  the  force 
F\  the  coefficient  of  friction  between  drawer  and  guides  being  0.2. 

Note.  —  KX.  p  the  guide  exerts  upon  the  drawer  the  force  H  and  another  force 

which  cannot  exceed  ////",   fi  being  the  coefficient  of  friction  between    drawer  and 

guides.     At  q  the  guide  exerts  upon  the  drawer,  a  force  G  and  another  force  fi  G.     Now 

when  it  is  just  possible  to  pull  out  the  drawer,  the  various  forces  F,  G^  H  and  the 

full  value  of  the  forces  fiH and  iiG  are  in  equilibrium.     The  algebraic 

conditions  of  equilibrium  are  : 

I.  That  the  sum  of  all  forces  to  the  right  be  equal  to  the  sum  of  all 
the  forces  to  the  left.     That  is 


H 


(i; 


2.   That  the  sum  of  all  the  downward  forces  be  equal  to  the  sum  of 
all  the  upward  forces.     That  is 


F=ixG-{-ixH 


(") 


Fig.  44/. 


3.  That  the  sum  of  all  the  right-handed  torque  actions  about  any 
chosen  origin  of  moments  be  equal  to  the  sum  of  all  the  left-handed  torque 
actions.  Choosing  the  point  q  as  the  origin  of  moments,  this  condition 
gives 

Right-handed  torques.         Left-handed  torques. 

yr(^_;^)=^  b^iH-^aH  (iii) 


in  which  a  is  the  depth  of  the  drawer  front  to  back,  and  b  is  its  width. 
All  three  unknown  forces  F,  G  and  H  may  be  eliminated  from  these 
equations  and  the  value  of  x  determined  in  terms  of  ^,  a  and  //. 

This  problem  should  be  solved  graphically  as  well  as  algebraically.  An  interest- 
ing modification  is  to  find  the  direction  of  F  when  x  is  given. 

44.  Given  a  tackle  block  arranged  as  shown  in  Fig.  44/.  Find 
the  weight  W  which  can  be  lifted  by  a  force  F  equal  to  150 
pounds-weight,  neglecting  friction. 

Note.  —  The  simplest  argument  of  this  problem  is  as  follows  :  The  tension  of  the 
rope  is  everywhere  equal  to  150  pounds-weight  if  friction  is  negligible.  Therefore, 
the  four  strands  of  rope  which  lead  to  the  lower  block  exert  a  total  lifting  force  of 
600-pounds  weight. 


CHAPTER  V. 

DYNAMICS.     TRANSLATORY   MOTION. 

27.  Force  and  its  effects.  —  Our  fundamental  notions  of  force 
arise  from  the  sensations  which  are  associated  with  muscular 
effort,  and  the  effects  of  force  are  extremely  varied.  Thus  a  force 
may  break  a  body  or  distort  it.  A  force  exerted  in  rubbing  one 
body  against  another  produces  heat.  When  ice  is  compressed 
it  melts,  when  vapor  is  compressed  it  condenses  to  liquid.  When 
an  unbalanced  force  acts  on  a  body  the  velocity  of  the  body 
changes.  The  fact  is  that  nearly  every  physical  phenomenon 
involves  force  action  of  one  kind  or  another. 

A  force  can  be  measured  only  in  terms  of  its  effects.  The 
effect  which  can  be  most  easily  used  for  the  measurement  of  force 
is  the  effect  the  force  has  in  distorting  a  body,  as,  for  example, 
in  stretching  a  helical  spring.  The  simplest  effect  of  a  force ^  how- 
ever, is  the  change  which  the  force,  when  unbalanced,  produces  in 
the  velocity  of  a  body,  inasmuch  as  this  effect  is  independejit  of  the 
nature  of  the  body.  This  effect  is  now  universally  adopted  as  the 
effect  by  which  a  force  is  measured. 

The  study  of  the  effects  of  unbalanced  forces  in  modifying  the 
motion  of  bodies  constitutes  the  science  of  dynamics. 

28.  Types  of  motion.  —  Motion  is  infinite  in  variety,  and  there 
are  certain  simple  types  of  motion,  the  discussion  of  which  consti- 
tutes the  science  of  mechanics.  Thus  we  have  translator y  mo- 
tion in  which  every  line  in  the  moving  body  remains  unchanged  in 
direction,  rotatory  motion  in  which  a  certain  line  in  the  moving 
body  remains  fixed,  oscillatory  motion  in  which  the  moving  body 
undergoes  periodic  changes  of  shape,  wave  m,otion  in  which  a 
localized  pulse  of  motion  travels  through  the  body,  and  simple 
motion  of  flow  of  fluids.  In  general,  a  body  may  not  only  per- 
form all  of  these  types  of  motion  simultaneously,  but  several 
varieties  of  each  type  may  coexist.     It  is  necessary,  however,  to 

5  65 


66  ELEMENTS   OF   MECHANICS. 

study  each  type  of  motion  by  itself,  and  some  help  is  afforded 
towards  the  keeping  of  the  several  types  of  motion  clearly  sepa- 
rated in  one's  mind  by  conceiving  of  an  ideal  body  which  can  per- 
form only  one  or  another  type  of  motion,  as  follows  : 

A  material  particle  is  an  ideal  body  so  small  that  the  only  sen- 
sible motion  of  which  it  is  capable  is  translatory  motion.  One 
must  not,  however,  lose  sight  of  the  fact  that  the  term,  material 
particle,  is  used  merely  to  rivet  one's  attention  to  translatory 
motion,  and  any  body  whatever  which  performs  translatory 
motion,  or  which  is  acted  upon  by  forces  which  tend  to  pro- 
duce translatory  motion  only,  may  be  thought  of  as  a  particle  if 
one  wishes  to  think  in  such  terms. 

A  rigid  body  is  an  ideal  body  which  cannot  alter  its  shape  and 
which  is  capable  only  of  translatory  motion  and  rotatory  motion. 
One  must  not,  however,  lose  sight  of  the  fact  that  the  term  rigid 
body  is  used  merely  to  exclude  the  idea  of  change  of  shape  in  the 
discussion  of  rotatory  motion. 

Bodies  assumed  to  possess  ideal  elastic  properties  are  useful  in 
describing  some  of  the  more  important  phenomena  of  oscillatory 
motion  and  wave  motion.  For  example,  the  isotropic  body  which 
has  the  same  properties  in  all  directions  ;  the  perfectly  elastic  body 
which  is  assumed  to  follow  Hooke's  law  exactly ;  the  perfectly 
flexible  string  which  is  useful  in  describing  the  oscillatory  motion 
of  strings,  and  so  on. 

An  incompressible  frictionless  fluid  is  an  ideal  fluid  which  can- 
not be  reduced  in  volume,  and  which,  once  in  motion,  would 
remain  in  motion  indefinitely.  The  idea  of  an  incompressible 
frictionless  fluid  is  useful  in  describing  some  of  the  more  im- 
portant aspects  of  fluid  motion. 

The  use  of  ideal  bodies  and  substances  in  the  development  of 
mechanics  may  seem  to  be  objectionable,  but  it  is  necessary  to  dis- 
cuss one  thing  at  a  time,  and  it  is  even  more  necessary  to  ignore 
the  interminable  array  of  minute  effects  which  always  accompany 
every  physical  phenomenon,  any  detailed  consideration  of  which 
would  complicate  every  engineering  problem  beyond  the  possi- 


DYNAMICS.     TRANSLATORY    MOTION. 


67 


bility  of  a  practical  solution.  Thus  one  may  describe  in  a  general 
way  the  motion  of  a  railway  train  along  a  straight  level  track  by 
specifying  its  velocity,  whereas  the  actual  motion  involves  the 
swaying  and  vibration  of  the  cars  and  the  rattling  of  every  loose 
part,  it  involves  a  complicated  phenomenon  of  motion  which  is 
called  journal  friction,  and  it  involves  the  yielding  of  the  track 
and  a  whirling,  eddying  motion  of  the  air,  it  is,  in  fact,  infinitely 
complicated,  and  the  railway  engineer  who,  for  example,  is  con- 
cerned merely  with  the  design  of  a  locomotive  of  adequate  power, 
sums  up  all  of  these  effects  in  a  rough  estimate  of  the  total  fac- 
tional drag  which  the  locomotive  has  to  overcome. 

29.  Types  of  force  action.  —  To  each  type  of  motion  there  is  a 
corresponding  type  of  force  action.  Thus  a  force  action  which 
tends  to  produce  translatory  motion  only,  is  called  a  linear  force^ 
and  a  force  action  which  tends  to  produce  rotatory  motion  is 
called  a  twisting  force  or  torque.  The  internal  force  actions  in  a 
distorted  body  which  is  oscillating  or  which  is  transmitting  wave 
motion,  are  resolvable  into  what  are  called 

hydrostatic  pressure,  longitudinal  stress,  and 
shearing  stress.  The  force  action  which 
tends-to  produce  the  obscure  motion  which 
constitutes  the  electric  current  is  called  elec- 
tromotive force.  This  mere  enumeration  of 
the  various  types  of  force  action  is  intended 
only  to  emphasize  the  classification  of  motion 
into  the  various  types  mentioned  in  the  pre- 
vious article. 

30.  Translatory  motion.  Center  of  mass. 
—  When  a  body  moves  so  that  every  line  in 
the  body  remains  unchanged  in  direction  the 
body  is  said  to  perform  translatory  motion, 
of  translatory  motion  is  the  motion  of  a  body  along  a  straight 
path,  either  with  constant  or  varying  velocity,  as  exemplified  by 
the  motion  of  a  car  along  a  straight  track  or  the  motion  of  a  ship 
on  a  straight  course.     The  most  general  case  of  translatory  motion, 


Bl 


K.' 


/' 


Fig.  24. 


The  simplest  case 


68        •  ELEMENTS   OF   MECHANICS. 

however,  is  where  a  given  point  of  a  body  describes  any  path 
whatever  in  any  way  whatever,  but  where  every  hne  in  the  body 
remains  unchanged  in  direction,  as  indicated  in  Fig.  24. 

Grasp  a  long  sHm  stick  at  its  middle  and  move  it  up  and  down 
and  to  and  fro  in  any  way,  but  without  changing  the  direction  of 
the  stick ;  it  seems,  with  the  eyes  closed,  as  if  the  stick  were  a 
heavy  body  concentrated  in  the  hand,  or  in  other  words  the  stick 
behaves  as  if  it  were  all  concentrated  at  its  middle  point.  That 
point  in  a  body  at  which  a  sijtgle  force  must  be  applied  to  pro- 
duce transiatory  motion  is  called  the  center  of  mass  *  of  the  body. 
The  center  of  mass  of  a  uniform  stick  is  at  the  middle  of  the  stick. 

31.  Displacement,  velocity,  and  acceleration  of  a  particle.  — 
When  a  particle  moves  from  one  position  to  another  it  is  said  to 
be  displaced  and  the  distance  (and  direction)  from  the  initial  to 
the  final  position  of  the  particle  is  called  the  displacement. 

The  displacement  of  a  particle  divided  by  the  time  during  which 
the  displacement  takes  place  is  called  the  average  velocity  of  the 
particle.  Inasmuch  as  the  particle  may  move  in  any  way  what- 
ever in  making  a  given  displacement,  it  is  evident  that  the  actual 
velocity  of  the  particle  at  successive  instants  during  the  displace- 
ment may  be  very  different  from  the  average  velocity ;  but  if  the 
interval  of  time  be  extremely  short  (and,  of  course,  the  displace- 
ment small)  then  all  irregularities  vanish,  in  accordance  with  the 
principle  of  continuity  as  stated  in  Art.  20,  and  therefore  the 
average  velocity  during  a  very  short  interval  of  time  is  the  actual 
velocity  of  the  particle  at  the  given  instant,  that  is,  during  the  very 
short  interval. 

When  the  velocity  of  a  particle  is  changing,  the  actual  change 
during  a  given  interval  of  time  divided  by  the  interval  is  called 
the  average  acceleration  of  the  particle  during  the  interval,  and  if 
the  interval  is  very  short  the  acceleration  so  defined  is  the  actual 
acceleration  at  the  given  instant. 

The  actual  velocity  of  a  particle  at  a  given  instant  is,  of  course, 
never  determined  by  any  attempt  to  observe  the  displacement 

*  Sometimes  called  center  of  gravity. 


DYNAMICS.     TRANSLATORY   MOTION.  69 

during  a  very  short  interval  of  time,  and  the  actual  acceleration 
of  a  particle  at  a  given  instant  is,  of  course,  never  determined  by 
any  attempt  to  observe  the  change  of  velocity  during  a  very  short 
interval  of  time.  The  only  importance  to  be  attached  to  the  above 
definitions  is  that  the  student  should  see  that  they  are  legitimate, 
so  that  the  ideas  may  be  used  intelligibly.* 
32.  Newton *s  laws  of  motion. 

I.  All  bodies  persevere  in  a  state  of  rest,  or  in  a  state  of  uni- 
form motion  in  a  straight  line,  except  insofar  as  they  are  made  to 
change  that  state  by  the  action  of  an  unbalanced  force. 

II.  (a)  The  acceleration  of  a  particle  is  parallel  and  propor- 
tional to  the  unbalanced  force  acting  on  the  particle. 

(d)  The  acceleration  which  is  produced  by  a  given  unbalanced 
force  is  inversely  proportional  to  the  mass  of  the  particle. 

III.  Action  is  equal  to  reaction  and  in  a  contrary  direction, 
(i)  The  first  law  describes  the  behavior  of  a  particle  upon  which 

no  unbalanced  force  acts.  The  behavior  is  simply  that  t/ie  velocity 
of  the  particle  does  not  change^  and,  conversely,  if  a  body  moves 
at  uniform  velocity  in  a  straight  line,  the  forces  which  act  upon  it 
are  balanced. 

(2')  The  second  law  describes  the  behavior  of  a  particle  when 
acted  upon  by  an  unbalanced  force.  The  behavior  is  simply  that 
the  particle  gains  velocity  in  the  direction  of  the  force  at  a  rate 
which  is  [a)  proportional  to  the  force  and  (b)  inversely  proportional 
to  the  mass  of  the  particle. 

When  Newton  expounded  this  fact  he  evidently  had  it  in  mind 
that  a  force  is  measured  by  some  effect  other  than  acceleration, 
otherwise  he  could  not  have  affirmed,  except  as  a  definition,  that 
the  acceleration  which  a  force  produces  is  proportional  to  the  force. 
The  production  of  acceleration  is  now  adopted  as  the  effect  by 
which  forces  are  measured. 

(2'')  The  second  law  may  be  further  paraphrased  as  follows  : 
[a)  The  amount  of  velocity  gained  by  a  given  particle  in  a  given 

*The  discussion  of  the  time  variation  of  velocity  which  is  given  in  Art.  21  should 
be  reviewed  at  this  point. 


70         .  .  ELEMENTS   OF   MECHANICS. 

interval  of  time  is  proportional  to  the  unbalanced  force  acting  on 
the  particle,  and  the  gained  velocity  is  parallel  to  the  force. 
(b)  The  amount  of  velocity  produced  by  a  given  unbalanced  force 
in  a  given  interval  of  time  is  inversely  proportional  to  the  mass 
of  the  particle  upon  which  the  force  acts,  and  the  gained  velocity 
is,  of  course,  parallel  to  the  force.* 

(3)  The  third  law  expresses  the  fact  that  a  force  is  always  due 
to  the  mutual  action  of  two  bodies,  that  this  mutual  action  always 
consists  of  a  pair  of  equal  and  opposite  forces,  and  that  one  of 
these  forces  acts  on  body  number  one  and  the  other  upon  body 
number  two.  The  mutual  force  action  between  two  bodies  is 
called  acHo7i  in  its  effect  upon  the  body  which  is  being  studied 
and  reaction  in  its  effect  upon  the  body^ which  is  not  being  par- 
ticularly studied,  in  the  same  way  that  a  trade  is  called  buying  as 
it  ^ects  one  person  or  selling  as  it  affects  the  other  person. 

Inertia.  —  That  property  of  a  particle  by  virtue  of  which  it  per- 
severes in  a  state  of  rest  or  in  a  state  of  uniform  motion  in  a 
straight  line  when  not  acted  upon  by  an  unbalanced  force  is 
called  inertia ;  and  the  word  inertia  is  generally  extended  in  its 
meaning  to  include,  not  only  this  passive  property,  but  also  the 
idea  of  reluctance  to  gain  velocity.  Thus  a  given  unbalanced  force 
would  have  to  act  for  a  longer  time  on  a  body  of  large  mass  than 
upon  a  body  of  small  mass  to  produce  a  given  velocity,  that  is, 
the  body  of  large  mass  has  the  greater  reluctance  to  gain  velocity, 
or  a  greater  inertia  in  the  extended  sense  of  that  term. 

^  An  extremely  fanciful  statement  of  the  second  law  of  motion  has  crept  into  some 
elementary  treatises  on  mechanics  as  follows:  "The  effect  of  a  force  is  the  same 
whether  it  act  alone  or  in  conjunction  with  other  forces,"  meaning,  of  course,  the 
accelerating  effect.  Now  acceleration  is  proportional  to  force  and  any  effect  which  is 
p7'oportional  to  a  cause  may  be  divided  into  parts  and  each  part  assigned  as  the  effect 
^f  a  corresponding  part  of  the  cause.  Thus,  if  the  results  of  the  labor  of  a  number 
of  men  are  proportional  to  the  number  of  men,  then  it  is  justifiable  from  physical 
considerations  to  give  each  man  an  equal  share  of  the  profits  ;  but  if  the  results  are 
not  proportional  to  the  number  of  men,  it  is  not  justifiable  physically  to  give  to  each 
man  an  equal  share  of  the  profits.  The  principle  of  dividing  cause  and  effect  into 
parts  which  correspond  each  to  each  when  cause  and  effect  are  proportional,  is  called 
the  principle  of  superposition  and  it  runs  through  the  whole  science  of  physics  and 
chemistry. 


DYNAMICS.     TRANSLATORY   MOTION.  /I 

33.  Units  of  force.    Formulation  of  the  second  law  of  motion,  — 

{a)  Dynamic  units  of  force.  —  Having  agreed  to  measure  a  force 
in  terms  of  its  effect  in  changing  the  velocity  of  a  particle,  we 
may  choose  as  our  unit  of  force  that  force  which,  acting  as  an 
unbalanced  force  on  unit  mass,  will  produce  unit  velocity  in  unit 
time  (unit  acceleration).  Thus  the  dyne  is  that  force  which, 
acting  for  one  second  as  an  unbalanced  force  on  a  one-gram 
particle,  will  produce  a  velocity  of  one  centimeter  per  second  (an 
acceleration  of  one  centimeter  per  second  per  second)  ;  and  the 
poundal .is  that- force  which,  acting  for  one  second  as  an  unbal- 
anced force  on  a  one-pound  particle,  will  produce  a  velocity  of 
one  foot  per  second  (an  acceleration  of  one  foot  per  second  per 
second).  The  dyne  is  the  c.g.s.  unit  of  force  and  it  is  much  used  ; 
the  poundal  is  seldom  used. 

Having  adopted  as  our  unit  of  force  that  force  which  will  pro- 
duce unit  acceleration  of  unit  mass,  it  is  evident,  from  Newton's 
second  law,  that  F  units  of  force  will  produce  F  units  of  accelera- 
tion of  a  particle  of  unit  mass,  or  Flm  units  of  acceleration  of  m 
units  of  mass ;  that  is,  FJm  is  eqzial  to  the  acceleration  a  which 
the  force  F  produces  or   Flm  =  a,    whence 

F=  ma  (3) 

in  which  F  is  the  value  of  an  unbalanced  force  in  dynes  (or 
poundals),  m  is  the  mass  of  a  particle  in  grams  (or  pounds),  and 
a  rs  the  acceleration  in  centimeters  per  second  per  second  (or  feet 
per  second  per  second). 

{U)  Weight.  Gravitational  units  of  force.  —  Let  ^  be  the  ac- 
celeration of  a  freely  falling  body  produced  by  the  unbalanced 
pull  of  the  earth,  let  m  be  the  mass  of  the  body,  and  let  W  be 
the  force,  expressed  in  dynamics  units,  with  which  the  earth  pulls 
the  body,  then,  according  to  equation  (3)  we  have 

W=mg  (4) 

in  which  W  is  the  weight  of  the  body  in  dynes  (or  poundals),  m 
is  its  mass  in  grams  (or  pounds),  and  g  is  the  acceleration  due  to 


72         •  ELEMENTS   OF   MECHANICS. 

gravity  expressed  in  centimeters  per  second  per  second  (or  in  feet 
per  second  per  second).  The  value  of  ^is  about  980  centimeters 
per  second  per  second  (or  32  feet  per  second  per  second),  so  that, 
according  to  equation  (4),  the  weight  of  a  gram  is  about  980 
dynes,  and  the  weight  of  a  pound  is  about  32  poundals. 

The  most  convenient  unit  of  force,  for  many  purposes,  is  the 
force  with  which  the  earth  pulls  the  unit  mass.  The  force  with 
which  the  earth  pulls  one  gram  is  called  the  gram-weight,  and  the 
force  with  which  the  earth  pulls  the  pound  is  called  the  pound- 
weight.  Thus  we  may  speak  of  5,000  pounds-weight,  meaning  the 
force  with  which  the  earth  pulls  a  mass  of  5,000  pounds. 

The  slight  variation  in  the  value  of  the  gram-weight  and  the 
pound-weight  at  different  places  on  the  earth  is  of  no  consequence 
in  those  cases  where  these  units  of  force  are  used.  Thus  the 
tensile  strength  of  a  given  grade  of  steel,  repeatedly  measured 
under  conditions  as  nearly  alike  as  it  is  possible  to  make  them, 
will  vary  from,  say,  100,000  to  105,000  pounds-weight  per  square 
inch,  that  is,  the  tensile  strength  of  a  given  grade  of  steel  is  in- 
herently indefinite  (like  the  length  of  an  angle  worm  !),  and  a 
variation  of  a  few  tenths  of  one  per  cent,  in  the  value  of  the  unit 
of  force  is  of  no  consequence  whatever. 

It  is  often  desirable,  in  discussing  practical  problems,  to  con- 
sider the  relation  between  force,  mass,  and  acceleration  when 
forces  are  expressed  in  pounds-weight,  mass  in  pounds,  and  ac- 
celeration in  feet  per  second  per  second.  In  this  case  equation 
(3)  becomes 

F=^^^nia  (5) 

inasmuch  as  the  unit  of  force  in  this  case,  namely,  one  pound- 
weight,  produces  an  acceleration  of  about  32  feet  per  second  per 
second  in  a  mass  of  one  pound. 

34.  Measurement  of  force.  —  {ci)  By  the  kinetic  method.  —  The 
force  (unbalanced)  acting  on  a  body  may  be  calculated  by  equa- 
tion (3),  the  mass  of  the  body  being  known  and  the  acceleration 
being  determined  by  observation.     This  method  for  measuring 


DYNAMICS.     TRANSLATORY   MOTION. 


73 


force  cannot  be  realized  in  its  simplicity,  but  it  forms  the  basis 
of  many  physical  measurements. 

(b)  By  the  counter-poise  method.  —  The  strengths  of  materials 
are  nearly  always  determined  by  applying,  as  the  breaking  force, 
the  weight  of  a  body  or  bodies  of  known  mass,  multiplied  in  a 
known  ratio  by  a  system  of  levers.  The  machine  for  carrying 
out  such  a  test  is  called  a  testing  machine  and  it  is  similar  in 
many  respects  to  the  ordinary  platform  balance-scale. 

(c)  By  means  of  the  spring  scale.  —  The  spring  scale  is  an 
arrangement  in  which  an  applied  force  stretches  a  spring  and 
moves  a  pointer  over  a  divided  scale.  The  movement  of  the 
pointer  is  proportional  to   the  force,  and,  the  movement  for  a 


Fig.  25. 

known  force  being  observed,  the  scale  can  be  divided  so  as  to 
read  the  value  of  any  force  directly.  The  use  of  the  spring- 
scale  is  exemplified  in  the  measurement  of  the  draw-bar  pull  of 
a  locomotive.  Figure  25  shows  a  scale  designed  for  this  purpose. 
The  blocks  A  A  are  rigidly  fixed  to  the  "  dynamometer  car"  and 
the  link  H  couples  with  the  locomotive.  A  pull  on  H  moves 
the  cross  bar  C  and  compresses  the  springs,  and  a  push  on  H 
moves  the  cross  bar  D  and  compresses  the  springs.  The  relative 
motion  of  E  and  B  actuates  a  pointer  which  plays  over  a  divided 
scale. 


74 


ELEMENTS   OF   MECHANICS. 


UNIFORMLY   ACCELERATED   TRANSLATORY   MOTION. 

35.  Falling  bodies.  —  When  a  constant  *  unbalanced  force  acts 
upon  a  particle,  the  particle  gains  velocity  at  a  constant  rate. 
Such  a  particle  is  said  to  perform  uniformly  accelerated  motion. 
A  body  falling  freely  under  the  action  of  the  constant  pull  of  the 
earth  is,  insofar  as  the  friction  of  the  air  is  negligible,  an  example 
of  uniformly  accelerated  motion. 

All  bodies  when  falling  freely  gain  velocity  at  the  same  rate,  air 
friction  being  ?iegligible.  Thus  two  bricks  together  fall  at  exactly 
the  same  increasing  speed  as  one  brick  alone.  The  doubled  pull 
of  the  earth  on  the  two  bricks  produces  the  same  acceleration  as 
the  single  pull  of  the  earth  on  one  brick.  Doubling  the  force 
and  doubling  the  mass  leaves  the  acceleration  unaltered. 

Consider  a  particle  which  gains  velocity  at  a  constant  rate  of 
g  centimeters  per  second  per  second,  a  falling  body  for  example. 
The  velocity  gained  in  /  seconds  is 


v=gt 


(0 


Let  v^  be  the  initial  velocity  of  the  particle.     Then    v^  -f  gt   is 
its  velocity  after  t  seconds,  and  its  average  f  velocity  during  the  / 

*  Constant  in  magnitude  and  unchanging  in  direction. 

t  Let  the  constantly  increasing  velocity  of  a  falling  body  be  represented  by  the 
ordinates  of  a  curve  of  which  the  abscissas  represent  elapsed  times.     The  "curve" 


Fie.  26. 

so  plotted  v^rill  be  a  straight  line   AB,    Fig.  26,  and  the  average  ordinate  of  any  por- 
tion  AB   of  this  line  is  equal  to   \{yx-\-  v^. 


DYNAMICS.     TRANSLATORY   MOTION. 


75 


seconds  is  J  [v^  +  (v^  +  g-t)]  or  z^^  +  J^/ ;  and  the  distance  d 
fallen  by  the  particle  during  the  /  seconds  is  equal  to  the  product 
of  the  average  velocity  into  the  time  t.     That  is, 

If  v^  is  zero,  equation  (ii)  becomes 

d=  \gt'  (iii) 

Eliminating  t  between  equations  (i)  and  (iii),  we  have 

2/  =  1/  2gd  (iv) 

which  expresses  the  velocity  of  a  body  after  it  has  fallen  a  dis- 
tance d. 

36.  Projectiles.  —  When  the  initial  velocity  v^  of  a  body  is 
zero  or  when  it  is  vertical,  we  have  the  ordinary  case  of  a  falling 
body,  and  equation  (ii)  of  Art.  3  5  can  be  solved  by  simple  arith- 
metic, the  only  complication  being  that  v^  is  to  be  considered 
negative  when  it  is  upwards.  When  the  initial  velocity  v^  is  not 
vertical,  as  in  the  case  of  a  tossed  ball,  the  falling  body  is  called 
a  projectile.     In  this  case  the  entire  argument  of  Art.  3  5   holds 


Fig.  27. 


Fig.  28. 


good  but  geometric  addition  must  be  substituted  for  arithmetic 
addition.  Thus  the  average  velocity  of  a  projectile  during  t 
seconds  is  equal  to  the  geometric  sum,  v^  -f-  \gt,  as  shown  in  Fig. 
27,  and  after  t  seconds  the  projectile  is  in  the  line  OB  at  a  dis- 
tance from  0  equal  to  t  times  the  numerical  value  of  the  average 
velocity.     Or  one  may  find  the  position  of  the  ball  after  t  sec- 


76 


ELEMENTS   OF   MECHANICS. 


onds  on  the  basis  of  equation  (ii),  considering  that  v^t  is  a  distance 
in  the  direction  of  v^,  that  \gf'  is  a  distance  vertically  downwards, 
and  that  the  sum  v^t  +  }^gf  is  a  geometric  addition  as  shown  in 
Fig.  28. 

The  orbit  of  a  projectile  is  a  parabola.  —  This  may  be  shown 
by  choosing  the  ;r-axis  of  reference  parallel  to  v^  and  the  j/-axis 
vertically  downwards.  Then  x  =  v^t  and  y  =  \gt'^^  whence,  by 
eliminating  t  we  have  the  equation  to  the  parabola. 

The  hodograph  to  the  orbit  of  a  projectile  is  a  vertical  straight 
line,  —  Draw  the  line  OPy  Fig.  29,  representing  the  velocity  of  a 

projectile  at  a  given  instant,  then, 
j         after    t     seconds,    the     vertical 
velocity  gt  will  be   gained,  and 
the  total  velocity  will  be  repre- 
sented by  OP' .     Therefore,  if  we 
imagine  the  line  OP  to  change  so 
as  to  become  OP'  after  /  seconds 
and  thus  represent  the  changing 
velocity  at  each  instant,  then  the 
end  P  will  move  vertically  down 
wards  at  a  constant  velocity. 
Range  of  a  projectile.  —  The  horizontal  distance  reached  by  a 
projectile  when  it  comes  to  the  level  of  the  gun  on  its  downward 
flight  is  called  the  range  of  the  projectile.     The  range  of  a  pro- 
jectile, ignoring  the  effects  of  air  friction,  is  given  by  the  equation 


Fig.  29. 


/=?!V 


sin  Q  cos  Q 


g 

in  which  v^  is  the  initial  velocity  of  the  projectile,  0  is  the  angle 
between  the  direction  of  v^  and  a  horizontal  line,  and  g  is  the 
acceleration  of  gravity.  This  expression  for  /  may  be  easily  de- 
rived with  the  help  of  the  relations  shown  in  Fig.  30,  namely, 


and 


/=  vJ  cos  6 


Igt^  —  v^t  sin  0 


DYNAMICS.     TRANSLATORY   MOTION. 


77 


37.  Effect  of  air  resistance  on  the  motion  of  a  projectile. — 

Bodies  which  are  projected  through  the  air  do  not  have  a  con- 
stant downward  acceleration,  because  of  the  resistance  which  the 
air  offers  to  their  motion,  and  therefore  the  simple  theory  of  pro- 
jectiles above  outlined  is  not  applicable  in  practice.  The  limita- 
tions of  this  simple  theory  may  be  stated  in  a  general  way  as 
follows  : 

id)  In  the  first  place  the  above  simple  theory  is  not  limited  to 
the  motion  of  an  ideal  particle.  The  pull  of  the  earth  upon  a 
projectile  tends  only  to  produce  translatory  motion  and  the  effect 


Fig.  30. 

of  the  pull  of  the  earth  is  the  same  whether  the  projected  body 
is  rotating  or  not,  or  whether  the  projected  body  is  oscillating  or 
not ;  the  center  of  mass  of  the  body  describes  in  every  case  a  smooch 
parabolic  curve  in  accordance  with  the  discussion  of  Art.  36. 
Thus,  if  an  iron  bar  is  thrown  through  the  air,  the  center  of  mass 
of  the  bar  describes  a  smooth  parabolic  orbit ;  or  if  the  bar  is  pro- 
jected by  hitting  it  a  sharp  blow  with  a  hammer,  the  center  of 
mass  of  the  bar  describes  a  smooth  parabolic  orbit  as  before. 
This  illustrates  a  very  important  extension  of  the  idea  of  a  ma- 
terial particle,  namely,  we  may  call  any  body  a  material  particle, 
zvhatever  the  character  of  its  motion  may  be,  the  idea  being  to 
direct  one's  attention  solely  to  that  part  of  the  motion  of  the 
body  which  is  translatory. 


78 


ELEMENTS   OF   MECHANICS. 


{b)  In  the  case  of  a  heavy  body  moving  slowly,  for  example, 
an  iron  ball  tossed  from  the  hand,  the  resisting  force  of  the  air  is 
very  small  com^red  with  the  weight  of  the  body,  and  the  motion 
of  the  body  approximates  very  closely  indeed  to  the  ideal  motion 
discussed  in  Art.  36. 

{c)  In  the  case  of  a  light  body,  or  in  the  case  of  a  heavy  body 


orbit  in  vacuum  ( parabola)j-aii^e  28  miles 


projected  at  high  velocity,  the  resisting  force  of  the  air  may  be 
very  large  so  that  the  motion  of  such  a  body  differs  widely  from 
the  ideal  motion  described  in  Art.  36.  Thus,  Fig.  31  shows  the 
actual  orbit  of  the  heavy  projectile  from  a  modern  high  power 


Fig.  32. 

gun,  and  the  dotted  line  shows  what  the  orbit  Avould  be  in  a 
vacuum. 

{d)  The  air  friction  on  a  rotating  projectile  generally  gives  rise 
to  a  force  which  pushes  the  projectile  side  wise.     This  side  force 


DYNAMICS.     TRANSLATORY   MOTION.  f9 

is  the  cause  of  the  curiously  curved  orbit  of  a  **  split-shot " 
tennis  ball,  and  of  a  base  ball  pitched  by  an  expert  pitcher.  The 
curved  arrows,  Fig.  32,  show^the  direction  of  rotation  of  a  base 
ball,  the  arrow  M  shows  the  direction  of  its  translatory  motion, 
the  arrow  F  shows  the  side  force  above  mentioned,  and  the  dotted 
curve  shows  the  curved  orbit.  The  cause  of  the  side  force  is 
that  the  very  rapid  motion  of  the  rough  surface  of  the  ball  at  a 
produces  an  accumulation  of  air  at  P,  whereas  there  is  a  less 
accumulation  of  air  at  /*',  and  the  excess  of  air  at  P  causes  the 
ball  to  glance  to  one  side,  thus  curving  the  orbit. 

TRANSLATORY  MOTION  IN  A  CIRCLE. 
38.  Velocity  and  acceleration  of  a  particle  moving  steadily  in  a 
circular  orbit.  —  Consider  a  particle  which  makes,  steadily,  n  rev- 
olutions per  second  in  a  circular  orbit  of  radius  r.  The  circum- 
ference of  the  orbit  is  27rr,  and,  inasmuch  as  the  particle  traverses 
the  circumference  n  times  per  second,  its  velocity  v  is 

V  =  2Trrn  (6) 

The  magnitude,  or  numerical  value,  of  the  velocity  v  is  con- 
stant ;  but  its  direction  is  changing  continuously,  this  continual 
change  of  direction  of  v  involves  acceleration,  and  the  state  of 
affairs  at  each  instant  during  the  steady  motion  of  a  particle  in  a 
circular  orbit,  is  most  clearly  shown  by  the  use  of  the  idea  of  the 
hodograph  as  explained  in  Art.  21.  It  is  instructive,  however, 
to  discuss  the  motion  of  a  particle  in  a  circular  orbit  without  ex- 
plicit reference  to  the  hodograph,  as  follows  : 

To  determine  the  acceleration  of  a  particle  which  is  moving 
steadily  in  a  circular  orbit,  it  is  necessary  to  consider  the  change 
of  velocity  during  a  very  short  interval  of  time.  The  circle, 
Fig.  33,  represents  the  orbit  of  the  particle,  and  at  a  given  instant 
the  particle  is  at  P.  At  this  instant  the  velocity  v^  of  the  particle 
is  at  right  angles  to  PO  and  it  is  represented  by  the  line  0' P' 
which  is  drawn  from  the  fixed  point  0\  After  the  small  lapse  of 
time  A/,  the  particle  will  have  moved  a  distance  z/  •  A/  to  the  point 


8o  ELEMENTS   OF   MECHANICS. 

Q,  and  its  velocity  will  be  v^,  which  is  represented  by  the  line  0'  Q\ 
The  change  of  velocity  Az/  is  evidently  parallel  to  PO  (or  to 
QO,  for  it  must  be  remembered  that  the  time  interval  A/  is  in- 
finitely small),  and,  since  the.  triangles  OPQ  and  O'P'  Q  are 
similar,  we  have 

Az/      V'  ^t 

in  which  v  is  written  for  the  common  numerical  value  of  v^  and 
v^,  and    V'^t   is  the  length  of  the  infinitesimal  arc   PQ   which  is 


Fig.  33. 

traversed  by  the  particle  during  the  time  interval  A/.  From 
equation  (i)  we  have 

A7  =  7  W 

but,  the  change  of  velocity  Az^  divided  by  the  time  interval  A^* 
during  which  the  change  takes  place  is  the  acceleration,  so  that, 
writing  a  for    ^vj^t,    equation  (ii)  becomes 

-=-  (7) 

The  direction  of  a  is,  of  course,  parallel  to  Av,  and  Av  is  par- 
allel to    PO.     Therefore  a  particle  zvhich  moves  steadily  in   a 


DYNAMICS.     TRANSLATORY   MOTION.  8 1 

circular  orbit  of  radius  r  has  a  steady  acceleration  towards  the 
center  of  the  circle^  and  this  acceleration  is  equal  to  v^^r^  where 
V  is  the  steady  velocity  of  the  particle. 

It  is  sometimes  convenient  to  have  a  expressed  in  terms  of  r 
and  n,  thus  we  may  substitute  the  value  of  v  from  equation  (6) 
into  equation  (7)  and  we  have 

a  =  ^TT^n^r  (8) 

Force  required  to  constrain  a  particle  to  a  circular  orbit.  —  When 
a  piece  of  metal  is  tied  to  a  string  and  twirled  in  a  circular  orbit 
the  string  pulls  steadily  on  the  piece  of  metal,  this  pull  of  the 
string  is  an  unbalanced  force  since  no  other  force  *  acts  on  the 
piece  of  metal,  and  the  value  of  the  force  in  dynamic  units  is  equal 
to  the  product  of  the  mass  of  the  particle  and  its  acceleration, 
according  to  equation  (3).  Therefore  we  may  substitute  the  value 
of  ^  from  equation  (7)  or  equation  (8)  in  equation  (3)  giving 

^=^t  (9) 

and 

F  =  ^.TT^n^rm  f  (10) 

where  F  is  the  force  in  dynamic  units  required  to  constrain  a 
particle  of  mass  m  to  3.  circular  orbit  of  radius  r,  v  is  the  velocity 
of  the  particle,  and  n  is  the  number  of  revolutions  per  second. 

39.  Examples  of  motion  in  a  circle.  —  {a)  A  one  pound  piece 
of  metal  twirled  five  revolutions  per  second  in  a  circle  four  feet  in 
radius  would,  according  to  equation  (10),  require  a  force  of  3,868 
poundals  or  about  1 2 1  pounds-weight  to  constrain  it  to  its  orbit. 

{b)  Each  particle  of  a  rotating  wheel  must  be  acted  upon  by  an 
unbalanced  force  to  constrain  the  particle  to  its  circular  path. 
If  we  consider  only  the  rim  of  the  wheel,  neglecting  the  effect  of 

*  Resistance  of  the  air  and  force  of  gravity  are  here  ignored. 

+  These  equations  express  7^  in  dynamic  units,  dynes  or  poundals  as  the  case  may 
be.  If  i^  is  to  be  expressed  in  pounds-weight  these  equations  become  F^^-^i^mv^jr 
and  F^=  -^j  {^.K^n^rm),  where  m  is  the  mass  in  pounds  of  the  moving  particle,  v  is 
its  velocity  in  feet  per  second,  r  is  the  radius  of  the  circle  in  feet,  and  n  is  the  number 
of  revolutions  per  second. 
6 


b 


82 


ELEMENTS    OF   MECHANICS. 


the  spokes,  it  is  evident  that  the  necessity  of  the  unbalanced  radial 
forces  gives  rise  to  a  state  of  tension  in  the  rim.  The  tension  in 
a  barrel  hoop  presses  each  portion  of  the  hoop  radially  against 
the  barrel  staves,  and  the  outward  push  of  the  staves  balances 
the  radial  force  due  to  the  tension  of  the  hoop  ;  but  the  tension 
in  the  rim  of  a  rotating  wheel  produces  an  unbalanced  radial 
force  on  each  particle  of  the  rim  which  force  produces  the  radial 
acceleration  which  is  associated  with  the  circular  motion. 

(c)  The  tension  of  a  belt  produces  a  radial  force  which  presses 
the  belt  radially  against  the  face  of  the  pulley.  When  the  belt 
and  pulley  are  in  motion,  however,  a  portion  of  the  belt  tension 
produces  the  radial  forces  required  to  constrain  the  particles  of 


Fig.  34-. 

the  belt  to  their  circular  paths ;  the  portion  of  the  belt  tension  so 
used  is  proportional  to  the  square  of  the  velocity  of  the  belt  and 
inversely  proportional  to  the  radius  of  the  pulley  (a  =  v^'/'r). 
Therefore,  belts  running  at  high  speeds  on  small  pulleys  have  a 
troublesome  tendency  to  slip,  unless  the  tension  is  very  great. 

(d)  The  centrifugal  drier  which  is  used  in  laundries  and  in 
sugar  refineries,  is  a  rotating  bowl  AB,  Fig.  34,  with  perforated 
sides,  in  which  the  material  MM  to  be  dried  is  placed.  The 
action  of  the  centrifugal  drier  may  be  clearly  understood  as  fol- 
lows :  Consider  two  solid  particles  a  and  ^,  Fig.  35,  with  a  drop 


DYNAMICS.     TRANSLATORY   MOTION.  83 

of  water  d  clinging  to  them.  Gravity  of  course  pulls  on  the  drop 
and  the  drop  adheres  to  a  and  b  so  that  the  particles  are  able  to 
exert  on  the  drop  a  force  F  sufficient  to  balance  gravity.  In  the 
centrifugal  drier,  however,  the  particles  would  have  to  exert  upon 
the  drop  a  force  equal  to  ^ir'nhni,  where  r  is  the  radius  of  the 
circular  path  described  by  the  particles  a  and  ^,  m  is  the  mass  of 
the  drop,  and  71  is  the  speed  of  the  drier  bowl  in  revolutions  per 
second,  and  this  force  Aprri^rin  may  be,  say,  1000  times  as  great 
as  the  weight  of  the  drop ;  but  the  drop  does  not  have  sufficient 
adherence  to  the  particles  to  enable  the  particles  to  hold  to  it  with 
so  great  a  force,  and  the  result  is  that  the  drop  is  not  constrained 
to  the  circular  path,  but  flies  off  tangentially.  The  action  of  the 
centrifugal  drier  is  as  if  a  piece  of  wet  cloth  were  jerked  so  quickly 
to  one  side  as  to  leave  the  water  behind. 

(e)  A  locomotive  on  a  railway  curve  describes  a  circular  path 
and  an  unbalanced  horizontal  force  (equal  to  mv^jr)  must  push 
the  locomotive  towards  the  center  of  the  curve  in  order  that  the 
locomotive  may  follow  the  curve,  and  of  course  this  horizontal 
force  must  be  exerted  on  the  locomotive  by  the  track.  It  is 
desirable,  however,  that  the  total  force  with  which  the  track 
pushes  on  the  locomotive  (which  is  equal  and  opposite  to  the  force 
with  which  the  locomotive  pushes  on  the  track)  shall  be  perpen- 
dicular to  the  plane  of  the  track,  and,  therefore,  the  outside  rail 
is  always  raised  on  a  railway  curve. 

Let  us  consider  the  proper  elevation  to  be  given  to  the  outside 
rail  when  the  velocity  v  of  the  locomotive  and  the  radius  r  of  the 
curve  are  given.  The  rails  are  shown  at  a  and  b  in  Fig.  36,  F'ls 
the  total  force  that  must  act  upon  the  locomotive,  and  the  angle  6 
is  the  required  elevation. 

The  vertical  component  of  i^is  what  sustains  the  locomotive 
against  gravity.  Therefore  this  vertical  component  is  equal  to 
ing  where  m  is  the  mass  of  the  locomotive  and  g  is  the  accelera- 
tion of  gravity.     That  is  : 

Fcos  6  =  mg  (i) 


84 


ELEMENTS   OF   MECHANICS. 


The  horizontal  component  of  F  is  the  unbalanced  force  which 
constrains  the  locomotive  to  its  circular  path.  Therefore  this 
horizontal  component  is  equal  to  7/;;?/^  according  to  equation 
(9).     That  is  : 


i^sin  e  = 


(ii) 


Therefore,  dividing  equation  (ii)  by  equation  (i),  member  by 
member,  we  have 

tan  ^  =  —  (iii) 

When  a  locomotive  is  traveling  on  a  curve  it  is  evident  that 
the  whole  locomotive  is  rotating  about  a  vertical  axis  at  such  an 
angular  speed  that  if  the  curve  were  a  complete  circle  the  loco- 


Fig.  36. 

motive  would  make  one  rotation  about  a  vertical  axis  every  time 
it  traversed  the  circular  curve.  Therefore  a  locomotive  ^travel- 
ing on  a  curve  does  not  perform  pure  translatory  motion ;  but 
here  again  is  an  instance  where  the  translatory  motion  may  be 
considered  by  itself,  for  as  long  as  the  locomotive  is  on  the  curve, 
its  rotating  motion  is  constant  and  introduces  no  complication. 


DYNAMICS.     TRANSLATORY   MOTION.  85 

If  a  locomotive  suddenly  enters  a  curve  from  a  straight  portion 
of  track,  however,  then  the  locomotive  would  have  to  change 
suddenly  from  zero  rotatory  motion  to  that  speed  of  rotatory 
motion  which  corresponds  to  the  curve,  and  the  rails  would  have 
to  exert  an  excessively  great  horizontal  force  on  the  front  wheels 
of  the  locomotive.  It  is  for  this  reason  necessary  to  avoid  sudden 
changes  of  curvature  of  a  railway  track.     Thus  Fig.  37  shows 


\o 


Fig.  37.  Fig.  38. 

a  straight  portion  of  a  track  changing  abruptly  to  a  circular  curve 
at  the  point  a,  and  Fig.  38  shows  the  same  straight  portion  chang- 
ing gradually  into  the  same  circular  curve. 

40.  The  rotating  hoop.  —  It  is  pointed  out  in  the  above  ex- 
amples of  circular  motion  that  the  radial  forces  which  constrain 
the  particles  of  the  rim  of  a  rotating  wheel  to  their  circular  paths 
are  (ignoring  effect  of  spokes)  due  to  a  state  of  tension  in  the 
rim.  The  tension  of  the  rim  is  the  force  F  with  which  any  por- 
tion of  the  rim  pulls  on  a  contiguous  portion.  Let  the  circles, 
Fig.  39,  represent  a  hoop  of'  a  radius  r  rotating  n  revolutions 
per  second  about  the  axis  C,  let  the  mass  per  unit  length  of  the 
rim  of  the  hoop  be  m\  and  let/  be  the  unbalanced  force  pull- 
ing radially  inwards  on  each  unit  length  of  the  rim  (due  to  the 
tension  in  the  rim).  Consider  a  very  short  portion  of  the  rim  of 
length  r  •  Ac/),  which  subtends  the  angle  A(/)  as  shown.  The  unbal- 
anced force  pulling  the  portion  r  ■  Acj)  radially  inwards  is/  x  r  •  A(^. 
Let  F^  be  the  force  with  which  portion  k  pulls  on  the  given 
portion  r  •  A^  at  the  point  a,  and  let  F^  be  the  force  with  which  the 
portion  /  pulls  on  the  given  portion  r  •  Ac^  at  the  point  d.  The 
forces  F^  and  F^  are  equal,  numerically,  and  they  are  tangent  to 


86 


ELEMENTS   OF   MECHANICS. 


the  circle  at  a  and  b  respectively.  Draw  the  two  lines  F^  and  F^, 
Fig.  40,  parallel  to  F^  and  F^,  Fig.  39,  and  complete  the  parallelo- 
gram of  which  F^  and  F^  are    the   sides.      Then    the  diagonal, 


Fig.  39. 


Fig.  40. 


/*x  r-A<(),  of  this  parallelogram  represents  the  total  unbalanced 
force  which  acts  on  the  given  portion  r  •  Ac/)  of  the  rim,  and  from 
the  similar  triangles  of  Fig.  39  and  Fig.  40  we  have 

rA(f)     /r-A<^ 
r      ~      F 

in  which  F'ls  written  for  the  common  numerical  value  of  F^^  and 
Therefore 


J'r 


f  = 


(0 


That  is,  the  unbalanced  inward  pull  on  each  unit  length  of  a 
hoop  is  equal  to  the  tension  of  the  hoop  divided  by  its  radius. 

Now  the  mass  of  unit  length  of  the  rim  is  equal  to  ni'\  and, 
according  to  equation  (10),  the  force  which  must  be  pulling 
radially  inwards  on  unit  length  of  the  nm  to  constrain  it  to  its 
circular  path,  is  equal  to  47r^;/V  times  its  mass   in' .     Therefore 


I 


DYNAMICS.     TRANSLATORY   MOTION.  8/ 

writing  ^ir't^rm'  for /in  equation  (i)  we  have 

F  —  Apr^f^r'm!  ( 1 1 ) 

in  which  F  is  the  tension  in  dynes  in  a  rim  r  centimeters  in  radius, 
rotating  n  revolutions  per  second,  and  m!  is  the  mass  in  grams 
of  one  centimeter  of  the  rim.  If  m'  is  expressed  in  pounds  mass 
per  foot  of  rim,  r  in  feet,  n  in  revolutions  per  second  and  F  in 
pounds-weight,  then 

Example. — The  rim  of  a  large  flywheel  has  a  mass  of  250 
pounds  per  foot,  the  radius  of  the  wheel  is  15  feet,  the  wheel 
rotates  one  revolution  per  second,  and  the  tension  of  the  rim 
(neglecting  the  effect  of  the  spokes)  is  69, 350  pounds-weight. 

TRANSLATORY    HARMONIC   MOTION.  l 

41.  Definition  of   harmonic   motion.     Utility  of   the   idea. — 

Simple  harmojiic  motion  is  the  projection  on  a  fixed  straight  line 
of  uniform  motion  in  a  circle.  Consider  a  point  P',  Fig.  41, 
moving  Uniformly  around  a  circle  of  radius  r  at  a  speed  of  n 
revolutions  per  second,  the  point  P,  which  is  the  projection  of  P' 
on  the  line  CD,  performs  simple  harmo7iic  motion. 

Vibration  or  cycle.  —  One  complete  up-and-down  movement  of 
the  point  P,  Fig.  41,  is  called  a  vibration  or  a  cycle. 

Frequency.  —  The  number  of  vibrations,  or  cycles,  per  second  is 
called  the  frequency  of  the  oscillations  of  the  point  P ;  this  is,  of 
course,  equal  to  the  number  of  revolutions  per  second  of  the 
point  P' . 

Period.  The  time  required  for  the  particle  to  complete  one 
whole  vibration,  or  cycle,  is  called  the  period  of  the  harmonic 
motion.  The  relation  between  the  frequency  n  and  the  period  t 
is  obviously 

'«  =  -  (12) 

Equilibrium  position.  When  the  vibrating  particle  P,  Fig.  41, 
is  at  the  point  0,  no  force  acts  upon  it,  as  explained  below ;  the 


88 


ELEMENTS   OF   MECHANICS. 


point  0  is  therefore  called  the  equilibrium  position  of  the  vibra- 
ting particle. 

Amplitude.  The  maximum  distance  from  0  reached  by  the 
vibrating  particle  is  called  the  amplitude  of  its  oscillations.  This 
amplitude  is  equal  to  the  radius  r  of  the  circle  in  Fig.  41. 

PJiase  difference.  Consider  two  points  P'  and  Q\  Fig.  42, 
both  making  n  revolutions  per  second  around  the  circle  so  that 


the  angle  Q  is  constant.  The  two  oscillating  particles  P  and  Q 
are  then  said  to  differ  in  phase  and  the  angle  0  is  called  their 
phase  difference. 

The  ideas  involved  in  the  peculiar  type  of  motion  which  is  per- 
formed by  the  particle  P,  in  Fig.  41,  are  used  throughout  the 
study  of  oscillatory  motion  and  wave  motion.  Thus  the  prongs 
of  a  vibrating  tuning  fork  perform  simple  harmonic  motion ;  the 
motion  of  a  pendulum  bob  is,  approximately,  simple  harmonic 
motion  ;  when  a  rod,  or  a  beam,  or  a  bridge  oscillates  in  the 
simplest  possible  manner,  each  particle  of  the  rod,  or  beam,  or 
bridge  performs  simple  harmonic  motion  ;  when  wave-motion  of 
the  simplest  kind  spreads  through  a  body  each  particle  of  the 
body  performs  simple  harmonic  motion. 


DYNAMICS.     TRANSLATORY   MOTION.  89 

An  example  of  simple  harmonic  motion  in  which  all  of  the 
details  of  Fig.  41  are  reproduced,  is  the  motion  of  the  cross-head 
of  a  steam  engine  with  a  long  connecting  rod.  The  crank  pin 
moves  at  sensibly  uniform  speed  in  a  circle,  one  component, 
only,  of  this  motion  is  transmitted  to  the  cross-head  by  the  long 
connecting  rod,  and  the  cross-head  moves  to  and  fro  in  the 
manner  of  the  point  P  in  Fig.  41. 

42.  Acceleration  of  a  particle  in  harmonic  motion.  The  veloc- 
ity of  the  point  P  in  Fig.  41  is  the  vertical  component  of  the 
velocity  of  the  point  P' ,  and  the  acceleration  a  of  the  point  P  is 
the  vertical  component  of  the  acceleration  a'  ^  therefore,  from  the 
similar  triangles  P' OP  and  P' cd  of  Fig.  41  we  have 

a       X 
a'  ~  r 

where  x  is  the  distance  OP,  and,  since  a'  =  ^.ir^n^r,  according 
to  equation  (8),  we  have 

^  =  —  ^irVx.  (13) 

The  minus  sign  is  introduced  for  the  reason  that  a  is  downwards 
(negative)  when  x  is  upwards  (positive). 

The  force  which  must  act  on  the  particle  P,  Fig.  41,  to  cause 
it  to  move  in  the  prescribed  manner,  is  at  each  instant  equal  to 
ma,    according  to  equation  (3),  therefore 

F=  —  /^ir^n^mx  (14) 

in  which  ;;/  is  the  mass  of  the  oscillating  particle,  x  is  the  distance 
of  the  particle  from  its  equilibrium  position  at  a  given  instant,  n 
is  the  frequency  of  the  oscillations,  and  F  is  the  force  which 
must  act  on  the  particle  at  the  given  instant. 

The  quantities  Jt  and  m  in  equation  (14)  are  constant.  There- 
fore equation  (14)  indicates  that  the  force  F  which  must  at  each 
instant  act  on  a  particle  in  harmonic  motion  is  proportional  to  the 
distance  x  of  the  particle  from  its  equilibrium  position,  that  is,  we 

may  write 

F^-kx  (15) 


90 


ELEMENTS   OF   MECHANICS. 


where 


k  =  /^irVm 


(i6) 


or  using  I  /t  for  n^  where  t  is  the  period  of  one  complete  oscilla- 
tion, we  have 


k  = 


4T^m 


(17) 


43.  Examples  of  the  application  of  equations  (15),  (16)  and 
(17).  —  (a)  Application  to  a  weight  attached  to  the  end  of  a  flat 
sprijtg.  A  weight  of  mass  in  is  fixed  to  one  end  of  a  flat  steel 
spring  S,  the  other  end  of  which  is  clamped  in  a  vise  as  shown 
in  Fig.  43.     If  the  weight  M  is  pushed  to  one  side  through  a 


if 


8 


Fig.  43.  Fig.  44. 

distance  Xy  the  spring  exerts  a  force  F  which  urges  the  weight 
back  towards  its  equilibrium  position  and  this  force  is  proportional 
to  X.     Therefore,  we  may  write 

F=-kx  (i) 

in  which  y^  is  a  constant,  the  value  of  which  may  be  determined 
by  observing  the  force  required  to  hold  the  weight  at  a  measured 
distance  x  from  its  equilibrium  position. 


DYNAMICS.     TRANSLATORY   MOTION.  9 1 

Now  since  the  equation  (i)  is  identical  to  equation  (15)  it  is 
evident  that  the  weight,  once  started,  will  perform  simple  harmonic 
motion. 

(b)  Application  to  the  simple  pendulum.  —  The  simple  pendulum 
is  an  ideal  pendulum  consisting  of  a  particle  /*,  Fig.  44,  suspended 
from  a  fixed  point  (9  by  a  string  /  of  which  the  mass  is  negligible. 
If  the  particle  P  is  moved  to  one  side  and  released  it  will  oscillate 
back  and  forth.  It  is  desired  to  shoiv  that  these  oscillations  are 
simple  harmonic  oscillations^  and  that  the  period  of  one  complete 
oscillation  is  equal  to  iirV  Ijg,  where  g  is  the  acceleration  of 
gravity. 

Let  Q  be  the  position  of  the  oscillating  particle  at  a  given  in- 
stant. The  length  x  of  the  circular  arc  PQ  is  equal  to  /<^,  and 
the  component  Qf  of  the  force  ing  with  which  gravity  pulls 
downwards  on  the  particle,  is  equal  to  mg  sin  <^  or,  if  the  angle 
(f>  is  very  small ^  this  force  is  equal  to  mg'<l>,  or  to  mg/l  times  <I>1, 
or  to  mg/l  times  x.  Therefore,  remembering  that  the  force 
Qf{=  F)  is  to  the  left  when  the  arc  PQ{=  x)  is  to  the  right,  we 
have 

r-        ^     X, 

But  this  equation  is  identical  in  form  to  equation  (15)  since  mgjl 
is  constant,  therefore  the  pendulum  bob  in  Fig.  44  performs  sim- 
ple harmonic  motion,  and  the  equation  expressing  the  period  of 
the  oscillations  may  be  found  by  substituting  mg\l  for  k  in 
equation  (17).     In  this  way  we  find 

mg      47r^m 
or 


=  ^^47 


(18) 


44.  Harmonic  motion  represented  by  a  curve  of  sines.  —  If  the 

point/",  Fig.  45,  moves  around  the  circle  at  uniform  speed,  the 
angle  P'  OA  is  proportional  to  elapsed  time,  and  it  may  be  writ- 


92 


ELEMENTS    OF   MECHANICS. 


ten  (at  where  «  is  a  constant  and  t  is  elapsed  time  reckoned  from 
the  instant  that  P'  was  at  A.     Therefore  we  may  write 

jr  =  r  sin  (ot.  ( 1 9) 

That  is,  the  distance  of  the 
vibrating  particle  P  from 
its  equilibrium  position  0 
is  proportional  to  the  sine 
of  a  uniformly  increasing 
angle,  and  if  values  of  x 
be  plotted  as  ordinates  and 
the  corresponding  values  of 
t  (or  (ai)  as  abscissas,  we 
will  have  a  curve  of  sines 
as  shown  in  Fig.  46.  If  a 
fine  pointer  be  attached  to 
the  prong  of  a  tuning  fork, 
the  pointer  may  be  made 
to  trace  a  curve  of  sines  by  setting  the  fork  in  vibration  and  draw- 
ing the  pointer  uniformly  across  a  piece  of  smoked  glass. 

SYSTEMS   OF   PARTICLES. 

45.  System  of  particles.  The  ideas  of  translatory  motion  may 
conceivably  be  extended  so  as  to  serve  as  a  basis  for  the  descrip- 
tion of  any  motion  of  any  body  or  substance,  by  looking  upon  the 
body  or  substance  as  a  collection  of  particles  and  considering  the 
varying  position,  velocity,  and  acceleration  of  each  particle.  A 
collection  of  particles  treated  in  this  way  is  called  a  system  of  par- 
ticles or  simply  a  system.  Thus  a  rotating  wheel  is  a  system  of 
particles,  a  portion  of  flowing  water  is  a  system  of  particles,  a 
given  amount  of  a  gas  is  a  system  of  particles.  The  word  con- 
figuration is  used  when  we  wish  to  refer  to  the  relative  positions 
of  the  particles  of  a  system  ;  thus  the  configuration  of  a  system 
is  said  to  change  when  the  particles  change  their  relative  positions. 

A  closed  system  is  a  system  no  particle  of  which  has  any  force 


DYNAMICS.     TRANSLATORY    MOTION. 


93 


acting  on  it  from  outside  the  system.  There  is  no  such  thing 
in  nature  as  a  closed  system,  but  the  conception  is  useful 
nevertheless. 

The  cases  in  which  it  is  not  only  conceivably  possible,  but 
actually  feasible  to  study  more  or  less  complicated  types  of 
motion  by  treating  the  moving  substance  as  a  system  of  particles, 
are  as  follows : 

(a)  The  case  in  which  the  system  consists  of  very  few  bodies 


sixis  of  time 


Fig.  46. 

and  where  each  body  may  be  treated  as  a  particle.*     This  case 
is  exempHfied  by  the  sun  and  planets. 

(b)  The  case  in  which  the  particles  of  a  system  move  in 
a  perfectly  regular  or  orderly  way.  Thus  the  particles  of  a 
rotating  wheel  move  in  an  orderly  fashion,  the  particles  in  a 
smoothly  flowing  liquid  move  in  an  orderly  fashion,  the  particles 
of  a  vibrating  string  move  in  an  orderly  fashion,  the  connected 
parts  of  any  machine  such  as  a  steam  engine  or  a  printing  press 

*0r  where  each  body  is  a  connected  system^  see  (^). 


94  ELEMENTS   OF   MECHANICS. 

move  in  an  orderly  fashion.  Any  system  in  which  orderly 
motion  takes  place  is  called  a  connected  system. 

(c)  The  case  in  which  the  particles  of  a  system  move  in  utter 
disorder,  without  any  connection  whatever  with  each  other.  In 
this  case  it  would  evidently  be  impossible  to  consider  the  actual 
motion  of  each  particle,  in  fact  the  only  possible  treatment  of 
such  a  system  is  a  treatment  based  on  the  idea  of  averages  and 
probable  departures  therefrom.  Thus  the  very  important  kinetic 
theory  of  gases  has  been  built  up  on  the  hypothesis  that  a 
gas  consists  of  innumerable  disconnected  particles  in  disordered 
motion. 

46.  Momentum.  —  In  the  discussion  of  a  system,  the  product 
mv  of  the  mass  of  a  particle  and  its  velocity  is  of  sufficient  im- 
portance to  warrant  its  receiving  a  name  ;  it  is  called  the  momen- 
tiini  of  the  particle,  it  is  a  vector  and  its  direction  is  the  same  as 
the  velocity  v  of  the  particle. 

When  an  unbalanced  force  acts  upon  a  particle,  of  course  the 
momentum  of  the  particle  changes ;  the  rate  of  change  of  the 
momentum  is  equal  aiid  parallel  to  the  force.  This  is  evident 
when  we  consider  that  a  change  of  velocity  ^v  means  a  change 
of  momentum  equal  to  m  •  Av,  which,  divided  by  the  elapsed  time 
A/,  gives  the  rate  of  change  of  momentum  ;  but  Av/At  is  equal 
to  acceleration,  so  that  ;;/  •  AvjAt  is  equal  to  mass  times  acceler- 
ation, and  this  is  equal  to  the  unbalanced  force,  according  to 
equation  (3),  Art.  33. 

The  mutual  foj^ce- action  of  two  particles  cannot  change  the  total 
momentum  of  the  two  particles.  This  is  evident  when  we  consider 
that  the  mutual  force-action  of  two  particles  consists  of  two  equal 
and  opposite  forces  (action  and  reaction),  so  that  while  one  par- 
ticle gains  momentum  in  one  direction,  the  other  particle  gains 
momentum  in  the  opposite  direction  at  the  same  rate.  The  con- 
stancy of  the  total  momentum  of  two  particles,  insofar  as  their 
mutual  force-action  is  concerned,  is  called  the  principle  of  the  con- 
servation of  momentum. 

The  principle  of  the  conservation  of  momentum  applies  to  any  number  of  particles 


I 


DYNAMICS.     TRANSLATORY   MOTION.  95 

insofar  as  their  n.  ""tual  force-actions  are  concerned.  The  total  momentum  of  the  par- 
ticles of  a  system  is.  never  changed  by  the  mutual  force-action  within  the  system,  or, 
in  other  words,  the  total  momentum  of  a  closed  system  is  constant. 

47.  Impact.  —  Consider  two  particles  of  which  the  masses  are  m^^  and  m^,  and 
the  velocities  v^  and  v^,  respectively.  The  combined  momentum  of  the  two  particles 
is  m^v^  -f-  m.fj^.  If  the  bodies  collide,  their  velocities  may  change, 
be  the  respective  velocities'after  impact.  Then  m^  V^  +  ^'^2  ^  is  th^ 
6f  the  bodies  after  impact,  and  by  the  principle  of  the  conservation  of  momentum,  we 
have 

Impact  of  inelastic  balls.  —  When  an  inelastic  ball,  such  as  a  ball  of  soft  clay 
strikes  squarely  against  another,  the  two  balls  move  after  impact  as  a  single  body  so 
that  V^  and  V.^  are  equal,  and  this  common  velocity  after  impact  is  completely  deter- 
mined by  equation  (i). 

Impact  of  perfectly  elastic  balls.  —  Consider  two  elastic  balls  moving  at  velocities 
v^  and  z/j  in  the  same  straight  line  {y^  and  v.j_  being  opposite  in  sign  if  the  balls  are 
moving  in  opposite  directions).     Let  the  masses  of  the  balls  be  m-^  and  m,^  respectively. 

When  these  balls  collide  they  are  distorted,  and  at  a  certain  instant  the  distortion 
reaches  a  maximum,  after  which  the  balls  rebound  from  each  other  and  the  distortion  is 
relieved.  When  the  distortion  of  the  two  balls  has  reached  its  maximum^  the  tzuo  balls 
are  at  the  instant  moving  at  common  velocity  r,  which  is  determined  by  the  equation 

(  m^  +  //Zj )  c  =  my^  +  Wj^j  ( ii) 

During  the  time  that  the  balls  are  being  distorted,  which  time  we  shall  call  the 
first  half  of  the  impact,  the  first  ball  loses*  an  amount  of  velocity  {y^  —  r)  and  the 
second  ball  loses*  an  amount  of  velocity  {v^  —  c).  During  the  time  that  the  balls 
are  being  relieved  from  distortion,  which  time  we  shall  call  the  second  half  of  the 
impact,  they  are  assumed  to  act  on  each  other  with  precisely  the  same  series  of  forces 
as  during  the  first  half  of  the  impact,  only  in  reverse  order.  This  is  what  is  meant 
by  the  assumption  that  the  two  balls  are  perfectly  elastic.  Therefore  during  the  sec- 
ond half  of  the  impact,  each  ball  loses  the  same  amount  of  velocity  as  it  lost  dur- 
ing the  first  half  of  the  impact,  that  is,  the  total  loss  of  velocity  by  the  first  ball  is 
2(z'j  —  r)  and  the  total  loss  of  velocity  by  the  second  ball  is  2(2/2  —  ^)'  ^°  ^^^ 

V^  =  v,-2(v,  —  c) 
and 

V^  =  V^—2{V,  —  C) 

or 

V^  =  2c — z/j  (iii) 

and 

f^  =  2<:  —  z/g  (i^) 

in  which  V^  and  V^  are  the  respective  velocities  of  the  balls  after  impact. 

*  The  velocity  c  lies  between  v^  and  v.^  so  that  if  (v^  —  c)  is  positive  then  (z/j  —  c) 
must  be  negative. 


96  ELEMENTS   OF   MECHANICS. 

Substituting  the  value  of  c  from  equation  (ii)  in  equations  (iii)  and  (iv)  we  have 


^  m^  -j-  7ft^  ^     ' 

The  simplest  case  is  where  m^  =  m^  and  where  v^  =  o,  that  is  where  the  balls  are 
similar,  and  where  the  first  ball  only  is  in  motion  before  impact.  In  this  case  the 
result  may  be  derived  from  equations  (v)  and  (vi)  but  it  is  more  instructive  to  derive 
the  result  anew.  The  common  velocity  c  at  the  middle  of  the  impact  is  equal  to  \v^. 
That  is,  the  first  ball  loses  half  its  velocity  and  the  second  ball  gains  an  equal  amount 
of  velocity  during  the  first  half  of  the  impact.  During  the  second  half  of  the  impact 
the  first  ball  loses  the  remainder  of  its  velocity  and  comes  to  a  standstill,  and  the  sec- 
ond ball  gains  once  more  an  equal  amount  of  velocity  so  that  its  velocity  is  now  equal 
to  the  initial  velocity  z\  of  the  first  ball.  That  is,  when  an  elastic  ball  A  strikes 
squarely  against  a  similar  stationary  ball  B,  the  ball  A  stops,  and  the  ball  £  moves 
on  with  the  full  original  velocity  of  A.  If  A  is  heavier  than  £,  then  both  balls  move 
in  the  same  direction  after  the  impact.  If  B  is  heavier  than  A,  then  A  moves  back- 
wards, or  has  a  negative  velocity  after  the  impact. 

48.  Motion  of  the  center  of  mass  of  a  system.  —  The  center  of 
mass  of  a  system  has  been  defined  in  physical  terms  in  Art.  25. 
The  center  of  mass  of  a  body  of  uniform  density  is  at  the  geo- 
metrical center  of  the  body.  The  center  of  mass  of  two  particles 
lies  on  the  line  joining  them,  and  its  distance  from  each  particle 
is  inversely  proportional  to  the  mass  of  the  particle.  Thus  the 
center  of  mass  of  the  earth  and  moon  is  oh  the  line  joining  the 
center  of  the  earth  and  the  center  of  the  moon,  and  it  is  about 
80  times  as  far  from  the  center  of  the  moon  as  it  is  from  the 
center  of  the  earth  (3,000  miles  from  the  center  of  the  earth), 
inasmuch  as  the  mass  of  the  earth  is  about  80  times  as  great  as 
the  mass  of  the  moon. 

The  ceftter  of  mass  of  a  system  remains  stationary,  or  continues 
to  move  with  uniform  velocity  in  a  straight  line,  if  the  vector  sum 
of  all  of  the  forces  zvhich  act  on  the  system  is  zero. 

For  example,  consider  an  emery  wheel  mounted  on  a  shaft  and 
rotating  at  high  speed.  If  the  center  of  mass  of  the  wheel  lies 
in  the  axis  of  the  shaft,  it  of  course  remains  stationary  as  the 
wheel  rotates,  and  the  only  force  that  need  be  exerted  on  the  shaft 
by  the  bearings  is  t»he  steady  upward  force  required  to  balance 


LCv. 


DYNAMICS.     TRANSLATORY   MOTION.  97 

the  downward  pull  of  the  earth  on  the  wheel.  A  rotating  ma- 
chine part  is  said  to  be  balaiiced  when  its  center  of  mass  is  in  its 
axis  of  rotation. 

When  the  center  of  mass  of  a  system  is  not  stationary^  and  does 
not  move  ivith  uniform  velocity  in  a  straigJit  line,  then  the  vector  sum 
F^  of  the  forces  which  act  on  the  system  is  not  zero. 

In  fact,  the  acceleration  A  of  the  center  of  mass  of  a  system  of 
particles,  the  vector  sum  F^  of  the  forces  which  act  on  the  system, 
and  the  total  mass  M  of  the  system  are  related  to  each  other 
precisely  in  the  same  way,  as  the  acceleration,  force,  and  mass  of 
a  single  particle.     That  is,  as  fully  explained  in  Art.  50,  we  have 

F^^  MA      ■  (20) 

Example  i.  —  Consider  an  emery  wheel  of  which  the  center  of 
mass  lies  at  a  distance  r  to  one  side  of  the  axis  of  rotation,  then, 
as  the  wheel  rotates,  the  center  of  mass  describes  a  circular  path 
of  radius  r,  the  acceleration  of  the  center  of  mass  is  equal  to 
Afirirr  at  each  instant,  and  a  side  force  equal  to  47rhi"rM  and 
parallel  to  r  at  each  instant  must  act  on  the  axle  to  constrain  the 
center  of  mass  to  its  circular  path,  precisely  as  if  the  entire  mass 
of  the  wheel  were  concentrated  at  its  center  of  mass. 

Example  2.  —  The  centrifugal  drier  consists  of  a  rapidly  rotat- 
ing bowl  mounted  on  top  of  a  vertical  spindle,  and  the  materials 
to  be  dried  are  placed  in  this  bowl.  It  is  impossible  to  keep  the 
bowl  and  contents  even  approximately  balanced,  so  that,  \{  the 
spindle  were  carried  in  a  rigid  bearing,  the  machine  would  be 
disabled  in  a  short  time  because  of  the  very  great  forces  that 
would  be  brought  into  play  in  constraining  the  center  of  mass  of 
bowl  and  contents  to  move  in  a  circular  path.  This  difficulty  is 
obviated  by  supporting  the  spindle  at  the  lower  end  only,  in  a 
long  bearing  mounted  on  springs  to  hold  it  approximately  vertical. 
The  bowl,  contents,  and  spindle  then  rotate  about  a  line  passing 
through  their  center  of  mass  and  through  the  center  of  the  flexi- 
ble bearing,  and,  although  the  bowl  and  spindle  seem  to  wobble 
badly  (inasmuch  as  they  do  not  rotate  about  the  axis  of  figure), 

7 


98      .  ELEMENTS   OF   MECHANICS. 

nevertheless  the  machine  runs  quite  smoothly,  producing  but 
little  vibration  in  the  supporting  frame. 

Example  j.  —  If  two  balls,  which  are  tied  together  with  a  short 
string,  are  thrown  in  such  a  way  that  the  string  is  kept  stretched 
while  the  balls  revolve  rapidly  about  one  another,  a  certain  point 
of  the  string  will  describe  a  smooth  parabolic  curve,  just  as  a 
simple  projectile  would  do.  This  point  of  the  string  is  the  center 
of  mass  of  the  two  balls.  The  center  of  mass  of  the  earth  and 
the  moon  describes  an  elliptic  orbit  about  the  sun  once  a  year, 
while  the  earth  and  moon  rotate  about  their  center  of  mass  once 
every  lunar  month,  in  a  manner  very  similar  to  the  motion  of  the 
two  balls  just  described. 

49.  The  balancing  of  a  rotating  machine  part.  Any  part  of  a 
machine  which  is  to  rotate  rapidly  must  be  adjusted  so  that  its 
center  of  mass  lies  in  the  axis  of  rotation.*  This  adjustment  is 
called  balancing,  and  a  machine  part  so  adjusted  is  said  to  be 
balanced.  A  machine  part  which  is  to  be  balanced,  a  dynamo 
armature  for  example,  is  mounted  on  its  shaft  and  the  ends 
of  the  shaft  are  placed  upon  two  straight  level  rails.  If  the 
center  of  mass  is  in  the  axis  of  the  shaft,  the  whole  will  stand  in 
equilibrium  in  any  position  ;  whereas,  if  the  center  of  mass  is  not 
in  the  axis  of  the  shaft,  the  whole  will  come  to  rest  with  the 
center  of  mass  at  the  lowest  possible  position,  and  material  must 
be  removed  from  one  side  until  the  center  of  mass  is  in  the  center 
of  the  shaft. 

Figure  47  shows  a  wheel  mounted  on  a  pair  of  balancing  rails. 
Such  a  pair  of  balancing  rails  is  a  prominent  feature  in  a  shop 
where  the  fly-wheels  of  large  engines  have  to  be  balanced. 

60.  Equations  of  center  of  mass.  The  position  of  the  center  of  mass  of  a  system 
of  particles  may  be  expressed  in  terms  of  the  positions  and  masses  of  all  of  the  par- 
ticles in  the  system  as  follows  :  Let  x  be  the  ^-coordinate  of  a  particle  whose  mass  is 
viy  let  x'  be  the  ;r-coordinate  of  a  particle  whose  mass  is  //z'',  let  x"  be  the  x-coordi- 

*  A  machine  part  which  is  long  in  the  direction  of  the  shaft  upon  which  it  rotates, 
may  have  its  center  of  mass  in  the  axis  of  the  shaft  and  yet  the  bearings  may  have 
to  exert  constraining  forces  upon  the  shaft  as  the  part  rotates.  A  long  cylinder 
loaded  on  opposite  sides  at  the  two  ends  is  an  example. 


DYNAMICS.     TRANSLATORY   MOTION. 


99 


nate  of  a  particle  whose  mass  is  m'^  and  so  on,  then  the  sum  mx  -|-  m^x'-\-  m"x'^-\  etc. 


divided  by  the  total  mass  of  the  system,   namely, 


m'' -\-  etc.,  gives  the 


^-coordinate  of  the  center  of  mass  of  the  system.     That  is,  the  a:-coordinate  of  the 
center  of  mass  is 

'^nix 


X  = 


Y.m 


(21) 


and  exactly  similar  expressions  may  be  formulated  for  the  ^-coordinate  and  for  the 
s-coordinate  of  the  center  of  mass. 

If  the  origin  of  coordinates  is  at  the  center  of  mass  of  the  system  then,  of  course, 
JSTis  equal  to  zero,  and  equation  (21)  becomes 


'Lmx 


(22) 


In  order  to  show  that  equation  (20)  is  true,  it  is  sufficient  to  consider  only  the  x- 
component  of  A^  and  the  jf-components  of  the  accelerations  of  the  respective  particles. 


Fig.  47. 

The  x-component  of  ^  is  d^XIdf^  and  the  x-components  of  the  accelerations  of  the 
respective  particles  are  d^xfdfl,  d^x^ jdt^,  d^x^^jdt^  and  so  on.  Therefore,  writing 
M  for  2;«  in  equation  (21),  and  differentiating  twice  with  respect  to  time,  we  have 


^d^X  d^x   ,       ,d^x^  ,  d^x'^  ,     ^ 

^-^  =  ^^  :7;f  +  ^'-^  + -TT^T  +  etc. 


dt^ 


dfi 


dfi 


but  m{d^xldfi)  is  the  x-component  of  the  force  acting  on  the  particle  m,  m^{d^x^  jdt^) 
is  the  ;f-component  of  the  force  acting  on  the  particle  m^  and  so  on,  so  that  the  right- 


lOO  ELEMENTS    OF   MECHANICS. 

hand  member  of  this  equation  is  the  sum  of  the  ;r-components  of  all  the  forces  acting 
on  the  particles  of  the  system,  and  this  is  equal  to  the  sum  of  the  ;»r-components  of  all 
of  the  external  forces  acting  on  the  particles  of  the  system  inasmuch  as  mutual  force- 
actions  between  the  particles  of  the  system  cancel  out  of  this  sum  because  such  mutual 
force-actions  consist  of  pairs  of  equal  and  opposite  forces.  Therefore,  the  right  hand 
member  of  the  above  equation  is  the  .r-component  of  the  total  external  force  Fs  which 
acts  on  the  system  and  the  above  equation  reduces  to 

^/ times  x-component  oi  A  =^  jc-component  oi  Fg  (i) 

and  we  may  show  in  exactly  the  same  way  that 

M  times  ^-component  of  A  =  j-component  of  F^  (ii) 

and 

3/ times  s-component  of  ^  =  s-component  of  Fg  (iii) 

These  three  equations  are  equivalent  exactly  to  the  single  vector  equation  (20). 

Problems. 

45.  A  train  having  a  mass  of  350  tons  (2,000  pounds)  starting 
from  rest  reaches  a  speed  of  50  miles  per  hour  in  2  i^  minutes. 
What  is  the  average  pull  of  the  locomotive  during  2^  minutes, 
dragging  forces  of  friction  being  neglected  ? 

46.  The  above  train  moving  at  a  speed  of  50  miles  per  hour  is 
brought  to  a  standstill  in  1 6  seconds  by  the  brakes.  What  is  the 
average  retarding  force  in  pounds-weight  due  to  the  brakes  ? 

47.  An  elevator  reaches  full  speed  of  8  feet  per  second  2^ 
seconds  after  starting.  With  what  average  force  in  pounds- 
weight  does  a  160-pound  man  push  down  on  the  floor  while  the 
elevator  is  starting  up  ?  The  elevator  is  stopped  (when  moving 
up  at  full  speed)  in  i  y^  seconds.  With  what  average  force  in 
pounds-weight  does  a  160-pound  man  push  down  on  the  floor 
while  the  elevator  is  stopping  ? 

Note.  —  In  the  first  case  the  upward  push  of  the  floor  on  the  man  exceeds  the 
weight  of  the  man  by  the  amount  which  is  necessary  to  produce  the  upward  accelera- 
tion ;  in  the  second  case  the  weight  of  the  man  exceeds  the  upward  push  of  the  floor 
by  the  amount  which  is  necessary  to  produce  the  downward  acceleration. 

48.  An  elevator  car  has  a  mass  of  1,000  pounds.  It  gains  a 
velocity  upwards  of  8  feet  per  second  in  2  ^  seconds  after  start- 
ing from  rest.  Calculate  {a)  the  tension  on  the  rope  while  the 
car  is  stationary,  {b)  the  average  tension  of  the  rope  while  the 


DYNAMICS.     TRANSLATORY   MOTION.  10 1 

car  is  starting  upward,  and  (c)  the  tension  of  the  rope  while  the 
car  is  moving  at  the  full  speed  of  8  feet  per  second. 

49.  A  train  having  a  mass  of  1,200  tons  (2,000  pounds)  is  to 
be  accelerated  at  }4  niile  per  hour  per  second  up  a  ^  per  cent, 
grade.  The  train  friction  is  10  pounds  per  ton.  Find  the  neces- 
sary draw-bar  pull  of  the  locomotive. 

A^o/<r.  —  A  ^  per  cent,  grade  is  one  that  rises  i  foot  in  200  feet  of  horizontal 
distance. 

50.  A  cord  is  strung  over  a  pulley.  At  one  end  of  the  cord 
is  a  10  pound  weight,  and  at  the  other  end  of  the  cord  is  a  1 1 
pound  weight.  Neglecting  the  weight  of  the  cord  and  the  fric- 
tion and  mass  of  the  pulley,  find  the  acceleration  of  each  weight 
and  the  tension  of  the  cord.  .    ,  „   ,  ,  ,  ,  - 

51.  A  falHng  ball  passes  a  given  point  at  a  Velocity  of  12  feet 
per  second.  How  far  below  the  point  is  th^e  bail  dLftejr|5.se;q9n<is:? 
How  far  does  the  ball  fall  during  the  fifth  second  after  passing 
the  given  point  ? 

52.  A  heavy  iron  ball  is  tossed  at  a  velocity  of  20  feet  per 
second  in  a  direction  30°  above  the  horizontal.  What  are  its 
horizontal  and  vertical  distances  from  the  starting  point  after  ^ 
second  ? 

JVo^e.  —  Find  vertical  and  horizontal  components  of  the  initial  velocity.  The  lat- 
ter component  remains  unchanged  while  the  vertical  motion  of  the  ball  is  precisely 
what  it  would  be  if  it  had  no  horizontal  motion. 

53.  A  heavy  shot  is  thrown  in  a  direction  30°  above  the  hori- 
zontal, it  strikes  the  ground  50  feet  from  the,  thrower,  and  the 
shot  is  5  ^  feet  above  the  ground  when  it  leaves  the  thrower's 
hand.     What  is  the  initial  velocity  7'  of  the  shot  ? 

A^o^e.  —  The  horizontal  velocity  v  cos  30°  is  constant,  the  time  of  flight  is 
i=  100  feet  -t-  {v  cos  30°),  and  5  feet  is  equal  to  v  sin  30°  X  ^+  z^^^^- 

54.  An  80-ton  (2,000  pounds)  locomotive  goes  round  a  rail- 
way curve  of  which  the  radius  is  600  feet  at  a  velocity  of  65  feet 
per  second.  With  what  force  in  pounds -weight  do  the  flanges 
of  the  wheels  of  the  locomotive  push  against  the  outer  rail  when 
the  outer  rail  is  not  elevated  ? 


I02  ELEMENTS   OF   MECHANICS. 

55.  Calculate  the  proper  elevation  to  be  given  to  the  outer  rail 
on  a  railway  curve  of  600  feet  radius  for  a  train  speed  of  65  feet 
per  second,  the  width  of  the  track  being  4  feet  8  ^  inches. 

56a.  The  tension  of  a  belt  is  50  pounds-weight.  With  what 
force  in  pounds-weight  does  the  belt  push  against  each  inch  of 
circumference  of  a  pulley  1 2  inches  in  diameter  when  the  pulley 
is  stationary  ? 

A/bU.  —  The  static  relation  between  tension  in  a  circular  hoop  and  actual  out- 
ward forces  acting  on  each  part  of  the  hoop  is  the  same  as  the  relation  between  ten- 
sion and  the  unbalanced  inward  forces  in  the  case  of  a  rotating  hoop  as  discussed  in 
Art.  40.     See  Art.  26  on  D'Alembert's  principle. 

56d.  The  mass  of  each  inch  of  length  of  the  belt  specified  in 
problem  56  is  i^  pound.  With  what  force  in  pounds-weight 
does  the  belt  j3u;sli  against  each  inch  of  circumference  of  the  1 2- 
inch  pulle}^  when  the  pulley  revolves  at  a  speed  of  1,500  revolu- 
tions per  HKiiute, 'tlie  actual  tension  of  the  belt  being  50  pounds- 
weight  ? 

57.  The  car  next  to  the  locomotive  in  a  train  is  35  feet  long 
between  bumpers  and  it  is  pulled  at  each  end  with  a  force  of 
10,000  pounds  (the  force  at  the  rear  end  of  the  car  is  of  course 
somewhat  less  than  the  force  at  the  front  end).  The  train  rounds 
a  circular  curve  of  1,000  feet  radius  at  a  speed  of  20  miles  per 
hour.  The  car  with  its  load  weighs  100,000  pounds.  Find  the 
horizontal  force,  at  right  angles  to  the  track,  with  which  the  track 
acts  on  the  car. 

JVofe. — The  portion  of  a  train  directly  behind  the  locomotive  is  under  tension  like  a 
belt,  and  the  tension  help3  to  constrain  the  cars  to  their  circular  path  exactly  as  in  the 
case  of  a  belt  passing  around  a  pulley.  In  solving  this  problem  it  is  sufficiently  accu- 
rate to  use  the  formula  jF^^  Tjr  in  which  Z'is  the  tension  ot  a  belt,  r  is  the  radius 
of  the  circular  arc  formed  by  the  belt,  and  i^is  the  radial  force  per  unit  length  of  belt 
due  to  T. 

58.  A  force  of  5x  lO^  dynes  deflects  the  end  of  the  spring  in 
Fig.  43  through  a  distance  of  1.25  centimeters.  What  is  the 
value  of  the  constant  k  in  equation  (15),  and  in  terms  of  what 
unit  is  this  constant  expressed  ?  How  much  force  would  be 
required  to  deflect  the  end  of  the  spring  through  a  distance  of  2 
centimeters  ? 


DYNAMICS.     TRANSLATORY   MOTION.  103 

59.  A  mass  of  2  kilograms  is  attached  to  the  end  of  the  spring 
specified  in  problem  58,  and  the  mass  is  set  vibrating.  How 
many  complete  vibrations  will  it  make  per  minute  ? 
•  60.  A  force  of  10  pounds-weight  deflects  the  end  of  the  spring 
in  Fig.  43  through  a  distance  of  0.02  foot.  What  is  the  value  of 
the  constant  k  in  equation  (15)  and  in  terms  of  what  unit  is  this 
constant  expressed?  A  mass  of  10  pounds  is  attached  to  the 
end  of  the  spring,  how  many  complete  vibrations  will  the  10 
pound  mass  make  per  minute  ? 

61.  What  is  the  length  /  of  a  simple  pendulum  which  makes 
one  complete  vibration  per  second  at  a  place  where  the  accelera- 
tion of  gravity  is  98 1  centimeters  per  second  per  second  ? 

62.  A  wheel  has  a  mass  of  50  pounds,  its  center  of  mass  is 
0.2  inch  from  the  axis  of  the  shaft  upon  which  the  wheel  rotates, 
and  the  speed  of  the  wheel  is  600  revolutions  per  minute.  How 
much  force  in  pounds-weight  must  act  on  the  shaft  to  constrain 
the  center  of  mass  to  its  circular  path  ?  What  is  the  direction  of 
the  force  at  each  instant  ? 

63.  A  ballistic  pendulum  AB^  Fig.  63/,  is  suspended  by  two 
cords  ss,  the  length  of  each  of  which 

is  400  centimeters,  and  the  body  AB  ^^ 

weighs  10  pounds.  A  rifle  bullet  of 
which  the  mass  is  0.005  pound,  strikes 
AB  at  the  point  indicated  by  the  short 

arrow,  and  the    velocity  imparted  to       ^3 

AB   carries    it   through  a   horizontal  A  B 

distance    of    8     inches    before    it    is  ^^' 

brought  to  rest  by  gravity.      Find  the  velocity  of  the  bullet.    The 
acceleration  of  gravity  is  32  feet  per  second  per  second. 

Note. — The  center  of  mass  of  AB  describes  the  arc  of  a  circle  of  which  the  radius 
is  /.  Calculate  the  vertical  movement  of  AB  from  the  known  value  of  /  and  the 
specified  horizontal  movement  of  AB.  Then  calculate  the  velocity  of  AB  which 
would  suffice  to  lift  AB  through  this  vertical  distance,  and  then  calculate  the  veloc- 
ity of  the  bullet  by  using  the  principle  of  the  conservation  of  momentum. 

64.  A  ball  weighing  550  pounds  is  shot  from  a  150,000-pound 


I04  ELEMENTS   OF   MECHANICS. 

gun  at  a  velocity  of  2,500  feet  per  second.  What  is  the  back- 
ward velocity  of  the  gun  as  the  ball  leaves  the  muzzle  ?  Sup- 
pose the  gun  is  allowed  to  move  back  two  feet  during  the  recoil, 
what  is  the  average  value  of  the  force  required  to  bring  it  to  rest  ? 

65.  An  ivory  ball  of  which  the  mass  is  500  grams,  and  of  which  the  velocity  is 
100  centimeters  per  second,  collides  with  a  stationary  ivory  ball  of  which  the  mass  is 
1,000  grams,  the  line  connecting  the  centers  of  the  balls  being  parallel  to  the  velocity 
of  the  moving  ball.  Find  the  common  velocity  of  both  balls  after  their  relative  mo- 
tion has  been  reduced  to  zero  during  the  first  half  of  the  impact,  and  find  the  velocity 
of  each  ball  after  impact ;  specify  direction  of  each  velocity. 

Note.  —  Assume  that  the  ivory  balls  are  perfectly  elastic  as  explained  in  Art.  47. 


CHAPTER  VI. 
FRICTION.     WORK  AND  ENERGY. 

51.  Friction.  —  A  body  in  motion  is  always  acted  upon  by 
dragging  forces  which  oppose  its  motion  and  tend  to  bring  it  to 
rest.     This  action  is  called  friction. 

Sliding  friction. — When  one  body  slides  on  another  the  motion 
is  opposed  by  a  frictional  drag.  Thus  the  cross-head  of  a  steam 
engine  slides  back  and  forth  on  the  guides,  a  rotating  shaft  slides 
in  its  bearings,  and  the  motion  is  in  each  case  opposed  by  a  fric- 
tional drag. 

Fluid  friction. —  The  flow  of  water  through  a  pipe  or  channel, 
the  motion  of  a  boat,  and  the  motion  of  a  projectile  through  the 
air  are  opposed  by  friction.  This  type  of  friction  is  called  fluid 
friction  and  it  is  discussed  in  a  subsequent  chapter. 

Rolling  friction. —  The  frictional  drag  upon  a  wheeled  vehicle  is 
due  in  part  to  the  sliding  friction  at  the  journals,  in  part  to  the 
friction  of  the  air,  and  in  part  to  the  continual  yielding  of  the  road 
or  track  under  the  wheels.  The  effect  of  this  yielding  is  very 
much  as  if  the  vehicle  were  continually  going  up  a  hill,  the  top 
of  which  is  never  reached.  The  frictional  drag  on  a  wheeled 
vehicle  due  to  the  yielding  of  the  road  or  track  is  sometimes 
called  rolling  friction. 

Frictional  drag  due  to  unevenness  of  a  road  bed. —  When  a 
vehicle  is  drawn  very  felowly  over  a  rough  road,  the  wheels  roll 
*'  up  hill,"  as  it  were,  when  they  strike  a  small  stone  and  then 
''down  hill "  again  when  they  leave  the  stone,  and  the  average 
value  of  the  pull  required  to  draw  the  vehicle  is  not  effected  by 
unevenness  of  road  bed  ;  but  if  the  speed  of  the  vehicle  is  in- 
creased, the  unevenness  of  the  road  bed  produces  a  very  consid- 
erable frictional  drag,  the  effect  is  as  if  the  wheels  were  being  all 
the  time  "  rolled  up"   a  succession  of  small  hills  not  to   "roll 

105 


io6 


ELEMENTS   OF   MECHANICS. 


down  "  again,  but  to  come  down  each  time  with  a  bump.  This 
kind  of  friction  shows  itself  in  the  vibration  and  swaying  of  a 
vehicle,  and  it  is  one  of  the  most  prominent  causes  of  frictional 
drag  upon  a  vehicle  which  is  driven  at  high  speed. 

52.  Coefficient  of  sliding  friction.  —  The  horizontal  force  H 
required  to  cause  a  body  to  slide  steadily  over  the  smooth  hori- 
zontal surface  of  another  body  is  approximately  proportional  to 
the  vertical  force  V  which  pushes  the  body  against  the  surface. 
That  is 

H=t,V        '-.    -"■  (23) 


in  which  Fis  the  force  with  which  a  body  is  pushed  against  any 
smooth  surface,  and  H  is  the  force,  parallel  to  the  surface,  which 
causes  the  body  to  slide.  The  proportionality  factor  /i  is  called 
the  coefficient  of  friction  ;  it  is  nearly  independent  of  the  contact 
area  of  the  sliding  substances  and  it  does  not  vary  greatly  with 
the  velocity  of  sliding.  Thus  the  coefficient  of  friction  of  wood 
on  a  smooth  metal  surface  is  about  0.40,  the  coefficient  of  fric- 
tion of  smooth  brass  on  smooth  steel  (not  oiled)  is  about  0.22. 


-V 


Fig.  48. 


Angle  of  friction.  Consider  a  block  B,  Fig.  48,  sliding  on  a 
table  TT  in  the  direction  of  the  dotted  arrow,  and  let  V  be  the 
force  with  which  the  block  is  pushed  against  the  table  and  H  the 
force  necessary  to  keep  the  block  in  motion.  Then,  since  the 
ratio  H\  V  is  constant  (that  is,  if  V  is  large,  H  is  large  in  pro- 
portion), it  is  evident  that  the  angle  <^  between  V  and  the  result- 


FRICTION.     WORK   AND    ENERGY.  lO/ 

ant  force  i^is  constant.  This  angle  is  called  the  angle  of  friction 
of  the  given  substances  B  and  T,  and,  evidently,  the  tangent 
of  (^  is  equal  to  the  coefficient  of  friction,  /t,  of  the  sliding 
substances. 

It  is  important  to  notice  that  the  force  F  in  Fig.  48  is  the  total 
force  which  the  sliding  block  exerts  upon  the  table.  The  block 
can  exert  on  the  table  any  force  whatever,  the  direction  of  which  lies 
inside  of  a  cone  described  by  rotating  the  line  F  about  V  as  an  axis, 
but  the  block  cannot  exert  on  the  table  a  force  the  direction  of  which 
lies  outside  of  this  cone.  This  statement  assumes  that  a  force  H 
which  is  sufficient  to  keep  the  block  B  sliding  is  sufficient  to 
start  it  sliding.  In  fact,  a  slightly  greater  force  is  required  to 
start  the  block  (not  on  account  of  acceleration,  but  because  of 
sticking)  than  is  required  to  keep  it  sliding. 

Nature  of  sliding  friction.  —  The  friction  between  two  sliding 
surfaces  is,  no  doubt,  due  in  part  to  a  continual  interlocking  and 
release  of  fine  protuberances  on  the  sliding  surfaces,  and  in  part 
to  a  continual  welding  together  and  tearing  apart  of  the  sub- 
stances as  they  come  into  intimate  contact.  When  the  sliding 
surfaces  are  distinctly  rough  there  is  no  regularity  whatever  in 
the  friction.  Surfaces  which  are  fairly  smooth,  however,  have  a 
well  defined  coefficient  of  friction,  especially  if  they  are  made  of 
unlike  materials.  Thus  wood  sliding  on  metal,  polished  steel 
sliding  on  brass  or  babbitt  metal,  hard  steel  sliding  upon  the  pol- 
ished surface  of  a  jewel,  all  have  fairly  well  defined  coefficients 
of  friction.  The  coefficient  of  friction  is  generally  small  in  value 
for  hard  polished  dissimilar  substances. 

Similar  substances  usually  have  a  large  coefficient  of  friction 
and  frequently  the  friction  is  very  irregular  between  similar  sub- 
stances. Thus  brass  on  brass  tends  to  weld  and  tear  in  a  most 
remarkable  manner,  and  a  clean  plate  of  glass  cannot  be  made  to 
slide  on  another  clean  glass  plate  at  all  (if  the  surfaces  are  very 
clean)  unless  there  is  an  air  cushion  between  them. 

53.  Active  forces  and  inactive  forces.  Definition  of  work. — 
Nothing  is  more  completely  established  by  experience  than  the 


I08  ELEMENTS   OF   MECHANICS. 

necessity  of  employing  an  active  agent  such  as  a  horse  or  a  steam 
engine  to  drive  the  machinery  of  a  mill  or  factory,  to  draw  a  car, 
or  to  propel  a  boat ;  and  although  the  immediate  purpose  of  the 
driving  force  may  be  described  in  each  case  by  saying  that  the 
driving  force  overcomes  or  balances  the  opposing  forces  of  fric- 
tion, still  the  fact  remains  that  the  operation  of  driving  a  machine 
or  propelHng  a  boat  involves  a  continued  effort  or  cost.  Indeed 
to  supply  a  man  with  the  thing  (energy)  which  will  drive  his  mill 
or  factory,  is  to  supply  him  with  a  commodity  as  real  as  the 
wheat  he  grinds  or  the  iron  which  he  fabricates  into  articles  of 
commerce.  Wheat  and  iron  are  sharply  defined  as  commodities 
in  the  popular  mind  on  the  basis  of  many  generations  of  com- 
mercial activity,  because  wheat  and  iron  can  be  stored  up  and 
taken  from  place  to  place,  and  because  change  of  ownership  is  so 
easily  accomplished  and  so  simply  accounted  for.  That  which 
serves  to  drive  a  mill  or  factory,  however,  cannot  be  stored  up 
except  to  a  very  limited  extent,  and  it  is  only  in  recent  years  that 
means  have  been  devised  for  transmitting  it  from  place  to  place 
and  that  an  exact  system  of  accounting  has  been  established  for 
governing  its  exchange.  A  clear  idea  of  energy  does  not  exist 
as  yet  in  the  popular  mind,  and  the  following  definitions  cannot 
be  expected  to  convey  a  full  and  clear  idea  at  once. 

The  common  feature  of  every  case  in  which  motion  is  main- 
tained is  that  a  force  is  exerted  upon  a  moving  body  and  in  the 
direction  in  which  the  body  moves.  Such  a  force  is  called  an 
active  force  *,  and  to  keep  up  an  active  force  requires  continuous 
effort  or  cost. 

A  force  which  acts  on  a  stationary  body,  on  the  other  hand, 
may  be  kept  up  indefinitely,  without  cost  or  effort ;  and  such  a 
force  is  called  an  inactive  force.  Thus  a  weight  resting  on  a 
table  continues  to  push  downward  on  the  table,  a  weight  sus- 
pended by  a  string  continues  to  pull   on  the  string,  the  main 

*  An  active  force  is  any  mutual  force  action  between  two  bodies  one  of  which 
moves  with  respect  to  the  other.  To  push  on  the  front  door  of  a  moving  car  is  not 
to  exert  an  active  force. 


FRICTION.     WORK   AND    ENERGY.  IO9 

spring  of  a  watch  will  continue  indefinitely  to  exert  a  force  upon 
the  wheels  of  the  watch  if  the  watch  is  stopped. 

The  idea  of  an  inactive  force  is  applicable  also  to  a  force  which 
acts  on  a  moving  body  but  at  right  angles  to  the  direction  in 
which  the  body  moves.  Thus  the  vertical  pull  of  the  earth  on  a 
railway  train  which  moves  along  a  level  track  is  an  inactive  force, 
the  forces  with  which  the  spokes  of  a  wheel  pull  on  the  moving 
rim  of  the  wheel  are  inactive  forces. 

An  active  force  is  said  to  do  work,  and  the  amount  of  work 

done  in  any  given  time  is  equal  to  the  product  of  the  force  and 

the  distance  that  the  body  has  moved  in  the  direction  of  the  force. 

That  is 

W=Fd  (24) 

in  which  F  is  the  force  acting  on  a  body,  and  Wis  the  work 
done  by  the  force  during  the  time  that  the  body  moves  a  dis- 
tance d  in  the  direction  of  K  If  d  is  not  parallel  to  F,  then 
IV  =  Fd  cos  6,  where  6  is  the  angle  between  F  and  d. 

Units  of  work.  The  unit  of  work  is  the  work  done  by  unit 
force  during  the  time  that  the  body  upon  which  the  force  acts 
moves  through  unit  distance  parallel  to  the  force. 

The  erg,  which  is  the  c.  g.  s.  unit  of  work,  is  the  work 
done  by  a  force  of  one  dyne  during  the  time  that  the  body  upon 
which  the  force  acts  moves  through  a  distance  of  one  centimeter 
in  the  direction  of  the  force.  The  erg  is,  for  most  purposes,  in- 
conveniently small,  and  a  multiple  of  this  unit,  \hQ  joule,  is  much 
used  in  practice.  The  joide  *  is  equal  to  ten  million  ergs, 
(10^  ergs). 

The  work  done  by  a  force  of  one  pound-weight  during  the 
time  that  the  body  upon  which  the  force  acts  moves  through  a  dis- 
tance of  one  foot  in  the  direction  of  the  force,  is  called  the  foot- 
pound.f 

*  It  is  frequently  convenient  to  have  a  name  for  that  unit  of  force  which  multiplied 
by  one  centimeter  gives  one  joule  of  work,  according  to  equation  (24).  This  unit 
of  force  may  be  called  the  joule  per  centimeter. 

f  The  kilogram-meter  \si}aQ.  viOxV  ^oxiG:  by  a  force  of  one  kilogram- weight  during 
the  time  that  the  body  upon  which  the  force  acts  moves  through  a  distance  of  one 


no  ELEMENTS   OF   MECHANICS. 

54.  Power.  —  The  rate  at  which  an  agent  does  work  is  called 
the  power  of  that  agent.  Thus  a  locomotive  exerts  a  pull  of 
15,000  pounds- weight  on  a  train  and  draws  the  train  through  a 
distance  of  500  feet  in  10  seconds.  The  work  done  is  7,500,000 
foot-pounds  which,  divided  by  the  time  interval  of  ten  seconds, 
gives  750,000  foot-pounds  per  second  as  the  rate  at  which  the 
locomotive  does  work. 

Units  of  power.  Power  may,  of  course,  be  expressed  in  ergs 
per  second,  in  joules  per  second,  or  in  foot-pounds  per  second. 
The  unit  of  power,  one  joule  per  second,  is  called  a  watt.  The 
horse-power^  which  is  extensively  used  by  engineers,  is  equal  to 
746  watts  or  to  550  foot-pounds  per  second. 

Power  developed  by  an  active  force.  —  Consider  a  force  F  acting 
upon  a  body  which  moves  in  the  direction  of  the  force  at  velocity 
V.  During  t  seconds  the  body  moves  through  the  distance  vt  and 
the  amount  of  work  done  is  -Fx  vt  according  to  equation  (24), 
and,  dividing  this  amount  of  work  by  the  time,  we  have 

P=Fv  (25) 

in  which  P  is  the  power  developed  by  an  active  force  F,  and  v  is 
the  velocity  with  which  the  body,  upon  which  F  acts,  moves  in 
the  direction  of  F.  If  F  is  expressed  in  dynes  and  v  in  centime- 
ters per  second,  then  P  is  expressed  in  ergs  per  second  ;  if  i^  is 
expressed  in  pounds-weight  and  v  in  feet  per  second,  then  P  is 
expressed  in  foot-pounds  per  second. 

Example.  —  A  horse  pulls  with  a  force  of  200  pounds  weight 
in  drawing  a  loaded  cart  at  a  velocity  of  3  feet  per  second  and 
develops  600  foot-pounds  per  second  of  power. 

Measurement  of  power.  —  Nearly  all  practical  measurements 
relating  to  work  are  measurements  of  power.  The  power  of  an 
agent  may  be  measured  as  follows  : 

{a)  The  value  of  an  active  force  and  the  velocity  of  the  body 

meter  in  the  direction  of  the  force.  The  foot-pound  unit  of  work  is  used  quite  gen- 
erally by  American  and  English  engineers,  and  the  kilogram-meter  unit  of  work  is 
used  in  those  countries  where  the  metric  system  has  been  adopted. 


FRICTION.     WORK   AND    ENERGY.  IH 

upon  which  the  force  acts,  may  be  measured  and  the  power  may- 
then  be  calculated  according  to  equation  (25). 

Examples.  —  ( i )  The  draw-bar  pull  of  a  passenger  locomotive 
is  measured  by  means  of  a  heavy  spring  scale  and  found  to  be 
6,000  pounds,  and  the  velocity  of  the  locomotive,  as  determined 
by  the  distance  traveled  in  a  given  time,  is  found  to  be  90  feet  per 
second.  From  these  data  the  net  power  developed  by  the  loco- 
motive (not  counting  the  power  required  to  propel  the  locomo- 
tive itself)  is  found  to  be  540,000  foot-pounds  per  second,  or 
991  horse-power. 

(2)  Let  a  be  the  area  in  square  inches  of  the  piston  of  a  steam 
engine,  let  p  be  the  average  steam  pressure  in  the  cylinder  in 
pounds  per  square  inch  as  measured  by  a  steam-engine  indicator, 
let  /  be  the  length  of  stroke  of  the  piston  in  feet,  and  let  n  be  the 
number  of  revolutions  per  second  made  by  the  engine.  Then 
the  average  force  pushing  on  the  piston  is  pa  pounds-weight,  and 
the  work  done  during  a  single  stroke  is  pxa  xl  foot-pounds, 
and  since  the  number  of  single  strokes  per  second  is  2n,  the 
power  developed  by  the  steam  \s palx  2n,  or  2paln  foot-pounds 
per  second.  The  power  of  an  engine  determined  in  this  way  is 
called  its  indicated  power,  to  distinguish  it  from  the  power  de- 
livered by  the  engine  to  the  machinery  which  it  drives.  The 
power  delivered  by  an  engine  is  always  less  than  its  indicated 
power  on  account  of  frictional  losses  in  the  engine. 

(3)  An  engine  to  be  tested  is  loaded  by  applying  a  brake  to  its 
flywheel ;  the  pull  on  the  brake  (reduced  to  the  circumference 
of  the  flywheel)  is  200  pounds-weight ;  the  velocity  of  the  cir- 
cumference of  the  flywheel,  as  determined  from  the  measured 
diameter  of  the  wheel  and  its  observed  speed  in  revolutions  per 
second,  is  80  feet  per  second  ;  and  the  power  developed  by  the 
engine  is  equal  to  200  pounds  x  80  feet  per  second,  which  is 
equal  to  16,000  foot-pounds  per  second  or  29  horse-power.  The 
power  of  an  engine  determined  in  this  way  is  called  its  brake  power. 

Figure  49  shows  the  arrangement  of  a  brake  for  measuring  the 
power  of  an  engine,  or  of  any  agent,  like  an  electric  motor  or 


12 


ELEMENTS   OF   MECHANICS. 


water  wheel,  which  dehvers  power  from  a  pulley.  The  spring 
scale  5  measures  the  force  at  the  end  of  the  brake  arm,  and  this 
observed  force  is  multiplied  by  ajr  to  find  the  equivalent  force 
at  the  surface  of  the  pulley,  where  a  is  the  length  of  the  arm  as 
shown  in  Fig.  49  and  r  is  the  radius  of  the  pulley. 

(b)  Power  is  frequently  measured  electrically.  Thus  the  power 
in  watts  delivered  by  a  direct-current  dynamo  is  equal  to  the 
product  of  the  electromotive  force  of  the  dynamo  in  volts  by  the 
current  in  amperes  delivered  by  the  dynamo. 

Power-time  units  of  work.  —  Inasmuch  as  nearly  all  practical 


Fig.  49. 

measurements  relating  to  work  are  measurements  of  power,  it 
has  come  about  that  a  given  amount  of  work  is  often  expressed 
as  the  product  of  power  and  time.  The  ivati-hour  is  the  amount 
of  work  done  in  one  hour  by  an  agent  which  does  work  at  the 
rate  of  one  watt,  the  kilowatt-hour  is  the  amount  of  work  done  in 
one  hour  by  an  agent  which  does  work  at  the  rate  of  one  kilowatt 
(one  kilowatt  is  1,000  watts),  and  the  horse-powcr-hoiir  is  the 
amount  of  work  done  in  one  hour  by  an  agent  which  does  work 
at  the  rate  of  one  horse-power. 

Efficiency.  —  The  efficiency  of  a  machine,  like  a  water  wheel, 


FRICTION.     WORK   AND    ENERGY.  II3 

a  steam  engine,  a  dynamo,  or  a  motor,  which  transforms  energy, 
is  defined  as  the  ratio  of  the  power  developed  by  the  machine  to 
the  total  power  delivered  to  the  machine. 

ENERGY. 

55.  Definition  of  energy.     Limits  of  the  present  discussion.  — 

Any  agent  which  is  able  to  do  work  is  said  to  possess  energy,  and 
the  amount  of  energy  an  agent  possesses  is  equal  to  the  total 
work  the  agent  can  do.  Thus  the  spring  of  a  clock  when  it  is 
wound  up  is  in  a  condition  to  do  a  definite  amount  of  work  and 
it  is  therefore  said  to  possess  a  definite  amount  of  energy. 

In  developing  the  idea  of  energy  it  is  important  to  distinguish 
between  an  agent  which  merely  transforms  energy  and  an  agent 
which  actually  has  within  itself  the  ability  to  do  a  certain  amount 
of  work.  Thus  the  steam  engine  merely  transforms  the  energy 
of  fuel  into  mechanical  work,  and  a  water  wheel  merely  trans- 
forms the  energy  of  an  elevated  store  of  water  into  mechanical 
work,  whereas  a  clock  spring  when  wound  up  has  a  store  of 
energy  within  itself 

Whenever  a  substance  or  a  system  of  substances  gives  up 
energy  which  it  has  in  store,  the  substance  or  system  of  substances 
always  undergoes  change.  Thus  the  fuel  which  supplies  the 
energy  to  a  steam  engine  and  the  food  which  supplies  the  energy 
to  a  horse,  undergo  chemical  change  ;  the  steam  which  carries  the 
energy  of  the  fuel  from  the  boiler  to  the  engine  cools  off  or  under- 
goes a  thermal  change  when  it  gives  up  its  energy  to  the  engine ; 
a  clock  spring  changes  its  shape  as  it  gives  off  energy ;  an  ele- 
vated store  of  water  changes  its  position  as  it  gives  off  energy ; 
the  heavy  fly  wheel  of  a  steam  engine  does  the  work  of  the 
engine  for  a  few  moments  after  the  steam  is  shut  off  and  the  fly 
wheel  changes  its  velocity  as  it  gives  off  its  energy. 

Not  only  does  a  substance  undergo  a  change  when  it  gives  up 

energy  by  doing  zvork,  but  a  substance  which  receives  energy  or 

has  work  done  upon  it  undergoes  a  change.     Thus  when  air  is 

compressed  by  a  bicycle  pump,  work  is  done  on  the  air  and  it 

8 


114      '  ELEMENTS   OF   MECHANICS. 

becomes  warm ;  the  work  done  in  keeping  up  the  motion  of  any 
machine  or  device  produces  heat  at  the  places  where  friction 
occurs ;  when  a  clock  spring  is  wound  up  it  stores  energy  by 
its  change  of  shape ;  when  water  is  pumped  into  an  elevated  tank 
it  stores  energy  by  its  change  of  position ;  a  large  part  of  the 
work  which  is  expended  on  a  heavy  railway  train  at  starting  is 
stored  in  the  train  by  its  change  of  velocity. 

We  are  now  facing  a  very  important  question  ;  shall  we  attempt 
a  complete  discussion  of  the  whole  theory  of  energy  at  once  by 
examining  into  all  kinds  of  changes  which  take  place  when  a 
substance  does  work  or  has  work  done  upon  it?  or  shall  we 
base  our  discussion  on  one  thing  at  a  time  ?  Most  assuredly  the 
latter.  Therefore  let  us  proceed  to  discuss  the  energy  relations 
involved  in  purely  mechanical  changes,  namely,  changes  of 
position,  changes  of  velocity,  and  changes  of  shape,*  and  let  us 
exclude  everything  else  from  our  present  discussion  such  as 
chemical  changes  and  thermal  changes. 

In  attempting  to  exclude  thermal  changes  from  our  present 
discussion,  however,  we  are  confronted  by  the  fact  that  friction 
(with  its  accompanying  thermal  changes)  is  always  in  evidence 
everywhere  ;  and  it  requires  a  very  high  degree  of  analytical 
power  to  think  only  of  purely  mechanical  changes  in  the  face  of 
such  a  fact.  This  necessary  feat  of  mental  effort  is  greatly  facili- 
tated by  the  use  of  the  idea  of  a  frictionless  system ;  and  this 
word  will  be  used  whenever  it  is  desired  to  direct  the  reader's 
attention  exclusively  to  the  energy  relations  that  are  involved  in 
purely  mechanical  changes. 

Before  proceeding  to  a  minute  examination  into  the  mechanical 
theory  of  energy,  it  is  desirable  to  establish  the  ideas  of  kinetic 
energy  and  potential  energy  on  the  basis  of  general  experience. 
Suppose  that  a  post,  standing  beside  a  railway  track,  is  to  be 
pulled  out  of  the  ground ;  can  a  car-load  of  stone  be  made  to  do 
the  work  ?  Certainly  it  can.  All  that  is  necessary  is  to  have 
the  car  moving  past  the  post  and  to  throw  over  the  post  a  loop 

*  Changes  of  shape  are  discussed  in  Chapter  VIII. 


FRICTION.     WORK   AND    ENERGY.  .  II 5 

of  cable  which  is  attached  to  the  moving  car.  A  moving  car  is 
able  to  do  work  ;  and  when  it  does  work  its  velocity  is  reduced, 
and  its  store  of  energy  decreased.  The  energy  which  a  body 
stores  by  virtue  of  its  velocity  is  called  the  kinetic  energy  of  the 
body. 

It  is  also  a  familiar  fact  that  a  weight  can  drive  a  clock,  but  in 
doing  so  the  position  of  the  weight  changes  and  its  store  of  en- 
ergy is  reduced.  The  energy  which  a  body  stores  by  virtue  of 
its  position  is  called  the  potential  energy  of  the  body. 

The  physical  reality  which  lies  behind  the  terms  kinetic  energy 
and  potential  energy  can  perhaps  be  shown  most  clearly  by  con- 
sidering a  bicycle  rider.  Suppose  that  the  rider  faces  a  steep  hill 
or  a  sandy  stretch  of  road  where  he  is  called  upon  to  do  an  un- 
usual amount  of  work.  Every  bicycle  rider  realizes  the  ad- 
vantage of  having  a  large  velocity  in  such  an  emergency.  This 
advantage  of  velocity  is  called  kinetic  energy.*  Or  suppose  that 
a  bicycle  rider  wishes  to  use  his  whole  strength,  or  more  if  he 
had  it,  in  covering  a  certain  distance ;  every  bicycle  rider  realizes 
the  advantage  of  being  on  top  of  a  hill  in  such  an  emergency ; 
this  advantage  of  position  is  called  potential  energy. 

56.  Kinetic  energy  of  a  particle.  —  The  kinetic  energy  of  a 
particle  is  given  by  the  equation 

W=\mv'  (26) 

in  which  W  is  the  kinetic  energy  in  ergs,  m  is  the  mass  of  the 
particle  in  grams,  and  v  is  its  velocity  in  centimeters  per  second. 
Proof  of  equation  {26).  The  kinetic  energy  of  a  particle  may 
not  only  be  defined  as  the  work  it  can  do  when  stopped,  but 
also  as  the  work  required  to  establish  its  motion.  Let  a  con- 
stant unbalanced  force  F  act  upon  a  particle  of  mass  m^  then 

F  =^  ma.  (i) 

*  Of  course  a  body  can  have  velocity  only  in  relation  to  another  body  and  the  idea 
of  kinetic  energy  is  an  idea  which  applies  strictly  to  a  system  of  particles  but  not  to 
an  individual  particle.     The  velocity  in  equation  (26)  is  velocity  referred  to  the  earth. 


Il6      .  ELEMENTS   OF   MECHANICS. 

After  /  seconds  the  velocity  gained  is 

V  =  at  (ii) 

and  the  distance  traveled  is 

d  =  laf.  (iii) 

Therefore,  multiplying  equations  (i)  and  (iii),  member  by  member, 
we  have 

Fd  =  ljnaH\  (iv) 

but  Fd  is  equal  to  the  work  done  on  the  particle  and  a'^t'^  is  equal 
to  v"^,  according  to  equation  (ii),  so  that  equation  (iv)  reduces  to 
IV—  ^7nv^. 

The  kinetic  energy  of  a  system  of  particles  is,  of  course,  equal 
to  the  sum  of  the  kinetic  energies  of  the  individual  particles  of 
the  system. 

When  mass  is  expressed  in  pounds  and  velocity  in  feet  per 
second,  then  the  kinetic  energy  of  a  particle  in  foot-pounds  is 
given,  approximately,  by  the  equation 

W^  ,\mv\  (27) 

57.  Potential  energy.  The  energy  stored  in  a  system  by  virtue 
of  the  configuration  of  the  system,  that  is,  by  virtue  of  the  rela- 
tive positions  of  the  parts  of  the  system,  is  called  the  potential 
energy  of  the  system.  For  example  a  weight  stores  energy  by 
virtue  of  its  position  relative  to  the  earth ;  a  bent  spring  stores 
energy  by  virtue  of  its  elastic  distortion. 

It  is  impossible  to  assign  a  definite  amount  of  potential  energy 
to  a  system  which  has  a  given  configuration,  for  it  is  impractic- 
able to  assign  a  definite  limiting  configuration  beyond  which  the 
system  cannot  go.  Thus  the  weight  of  a  clock  might  have  its 
available  store  of  potential  energy  increased  by  boring  a  hole  in 
the  clock  case  so  that  the  weight  could  move  down  to  the  floor, 
then  a  hole  could  be  bored  in  the  floor  and  eventually  a  deep 
well  could  be  dug  in  the  ground.  In  order  to  be  able  to  speak 
definitely  of  the  potential  energy  of  a  weight  it  is  necessary,  there- 


FRICTION.     WORK   AND    ENERGY. 


117 


given  position 


fore,  to  assign  an  arbitrary  ^ero  position  and  to  reckon  the  potential 
energy  of  any  other  given  position  as  the  work  the  weight  can  do 
in  changing  from  the  given  position  to  the  chosen  zero  position. 

In  general  the  potential  energy  of  any  system  in  a  given  con- 
figuration may  be  defined  as  the  amount  of  work  the  system  can 
do  in  changing  from  the  given  configuration  to  an  arbitrarily 
chosen  zero  configuration. 

Conservative  systems.  —  A  system  (frictionless)  which  does  the 
same  amount  of  work  in  passing  from  one  configuration  to  an- 
other, whatever  the  intermediate  stages  may  be  through  which 
the  system  passes,  is  called  a  conservative  system,  and  the  idea  of 
potential  energy  applies  only  to  such  systems.  Suppose,  for  ex- 
ample, that  a  weight  would  do  more  work  in  moving  from  a 
given  position  to  its  chosen  zero  position  over  one  path  A  than 
over  another  path  B,  see  Fig.  ^oa ;  then  the  potential  energy  of 
the  weight  in  the  given  position 
would  be  indefinite  ;  and  if  the 
weight  were  carried  around  the 
closed  path  AB  in  the  direction  of 
the  arrows,  then  a  large  amount  of 
work  would  be  done  in  passing 
down  path  A  and  only  a  portion  of 
this  work  would  be  required  to 
carry  the  weight  back  to  the  given 
position  over  path  B.    That  is,  work 

would  be  created  eveiy  time  the  weight  completed  the  cycle  o^ 
motion  around  AB,  and  we  would  have  "perpetual  motion,"  that 
is  a  machine  which  could  do  work  without  suffering  any  perma- 
nent change  of  any  kind.  *  All  physical  systems  are  conservative 
insofar  as  purely  mechanical  changes  are  concerned  ;  and  experience 
shows  that  all  physical  systems  are  conservative  when  changes 

*  The  idea  involved  int  his  discussion  of  Fig.  50a  may  be  strengthened  by  introduc- 
ing the  idea  of  cheating  to  which  it  stands  in  clear  apposition.  Suppose  one  were  to 
hold  a  weight  in  his  hand  and  allow  it  to  move  downwards  in  full  view  of  a  class, 
and  then  bring  it  again  to  its  former  position  by  passing  it  behind  his  back  where  it  is 
out  of  sight  with  the  idea  of  avoiding  the  doing  of  work  ! 


ii8 


ELEMENTS   OF   MECHANICS. 


of  all  kinds,  mechanical,  chemical,  thermal  and  electrical,  are 
taken  into  account ;  that  is,  the  energy  that  a  system  gives  off 
when  it  undergoes  any  change  whatever,  depends  only  upon  the 
initial  and  final  states  of  the  system,  and  is  independent  of  the 
intermediate  stages  through  which  the  system  may  be  ,made 
to  pass. 

Perpetual  motion  impossible.  —  A  perpetual  motion  machine 
would  be  a  device  which  would  furnish  a  continuous  supply  of 
energy  for  driving  machinery.  Most  of  the  attempts  to  produce 
perpetual  motion  have  been  quite  ridiculous,  but  on  the  other 


Fig.  50-5.  Fig.  50c. 

hand  many  attempts  have  been  quite  reasonable.  The  reason- 
able attempts  have  nearly  all  been  attempts  to  get  more  work  out 
of  a  weight  while  it  falls  along  one  path  than  is  required  to  carry 
the  weight  back  to  its  starting  point  along  another  path.  Figures 
50^  and  50<f  show  two  perpetual  motion  devices  which  were  pro- 
posed and  tried  about  1750.  Figure  50^  is  a  rachet  wheel  to 
which  a  number  of  hinged  arms  are  attached,  each  arm  carrying 
a  heavy  weight.  Figure  5 or  is  a  wheel  to  the  rim  of  which  a 
number  of  bent  tubes  are  attached,  each  tube  containing  mercury. 
The  arrows  show  the  directions  in  which  the  wheels  were  expected 
to  be  driven  by  the  increased  leverage  of  the  falling  weights  or 
of  the  falling  mercury. 


FRICTION.     WORK   AND    ENERGY.  I  19 

58.  Mutual  relation  between  the  kinetic  energy  and  the  poten- 
tial energy  of  a  closed  system.  —  A  closed  system  is  a  system 
upon  which  no  outside  forces  act,  or,  in  other  words,  it  is  a  sys- 
tem which  neither  gives  off  nor  receives  energy.  Such  a  system 
does  not  exist  in  nature,  but  the  most  clearly  intelligible  state- 
ment of  the  idea  of  the  impossibility  of  perpetual  motion  may  be 
made  by  referring  to  such  an  ideal  system. 

Suppose  that  the  particles  of  a  closed  system  are  in  motion  and 
let  us  consider  what  takes  place  in  any  very  short  interval  of 
time.  In  the  first  place,  each  particle  moves  through  a  certain 
small  distance,  the  configuration  of  the  system  is  changed  ac- 
cordingly, and  the  potential  energy  of  the  system  decreases  by 
an  amount  which  is  equal  to  the  total  work  done  on  all  of  the 
particles  by  their  mutual  force  actions.  In  the  second  place,  the 
kinetic  energy  of  each  particle  increases  by  an  amount  equal  to 
the  work  done  upon  it,  and,  of  course,  the  total  kinetic  energy  of 
the  system  increases  by  an  amount  which  is  equal  to  the  total 
work  done  on  all  of  the  particles  by  their  mutual  force  actions. 
Therefore,  the  decrease  (or  increase)  of  potential  energy  of  a 
closed  system  is  always  equal  to  the  accompanying  increase  (or 
decrease)  of  the  kinetic  energy  of  the  system,  or,  in  other  words, 
the  sum  of  the  potential  and  kinetic  energies  of  a  closed  system 
is  constant. 

59.  The  principle  of  the  conservation  of  energy. — The  argu- 
ment of  Art.  58  is  based  upon  the  simplest  aspect  of  Newton's 
laws  of  motion,  and  the  conclusion  reached,  namely,  the  constancy 
of  the  total  energy  of  a  closed  system,  follows  directly  from  the 
idea  of  a  conservative  system  as  developed  in  Art.  57,  that  is, 
from  the  idea  of  potential  energy.  The  principle  of  the  con- 
servation of  energy  reduced  to  its  simplest  terms  is  that  the  work 
done  by  a  system  depends  only  upon  the  initial  and  final  states  of 
the  system  and  it  is  hopeless  to  seek  a  roundabout  method  for 
bringing  the  system  back  to  its  initial  state  by  a  smaller  expendi- 
ture of  work. 

The  usual  statement  of  the  principle  of  the  conservation  of 


120 


ELEMENTS   OF   MECHANICS. 


energy  is  that  energy  can  be  neither  created  nor  destroyed ;  but  this 
statement  is  so  completely  abstracted  from  actual  physical  con- 
siderations, that  it  is  almost  meaningless. 

60.  The  application  of  the  principle  of  work  to  statics.  The 
principle  of  virtual  work.  —  Consider  a  body  which  is  acted  upon 
by  any  number  of  forces ;  then,  if  the  body  should  be  given  any 
slight  displacement  whatever,  the  total  work  which  would  be  done 
by  all  the  forces  is  called  the  virtual  work,  and  this  work  is  equal 
to  zero  when  the  forces  are  in  equilibrium. 

This  principle  of  virtual  work  furnishes  the  simplest  basis  for 
formulating  the  conditions  of  equilibrium  in  many  complicated 
mechanisms. 


ko 


b 


Fig.  51. 

Example  i.  Let  the  body  shown  in  Fig.  5 1  be  turned  about  the 
point  0  through  the  angle  A</)  causing  the  point  of  application  of 
the  force  A  to  move  downwards  through  the  distance  a  •  A^  and 
the  point  of  application  of  the  force  B  to  move  upwards  through 
the  distance  b  •  Acf) ;  then  Aa  •  Ac^  is  the  work  do7ie  on  the  body 
by  the  force  A,  and  Bb  A(j>  is  the  work  done  by  the  body  on 
the  agent  which  exerts  the  force  B.  Therefore  Bb  •  A0  is  to 
be  considered  as  negative  work  done  on  the  given  body,  so  that 
according  to  the  above  principle  we  have 

Aa-A<j>  -\-  Bb-A(t)  =  o 


or 


Aa  -Jf  Bb  =  o 


FRICTION.     WORK   AND    ENERGY. 


121 


which  shows  the  relation  between  the  two  forces  which  must  be 
satisfied  if  their  combined  tendency  to  turn  the  body  about  the 
point  0  is  zero. 

Exmnple  2.  The  following  discussion  of  the  tension  in  a 
rotating  hoop  furnishes  a  good  example  of  the  use  of  D'Alem- 
bert's  principle  (see  Art.  26)  and  of  the  principle  of  virtual  work. 

Let  7n'  be  the  mass  per  unit  length  of  circumference  of  a 
circular  hoop  of  radius  r  rotating  about  its  axis  of  figure  at  a 
speed  of  ;/  revolutions  per  second.  The  radial  acceleration 
of  each  particle  of  the  hoop  is  ^ir-n^i^  according  to  equation  (8), 
so  that  each  unit  of  length  of  circumference  u  of  the  hoop  must 
be  pulled  inwards  by  an  unbalanced  force  /  equal  to  ^ir^i^rm'^ 
according  to  equation  (10)  as  shown  in  Fig.  52^. 


Fig.  52a. 


Fie.  52^. 


By  applying  D'Alembert's  principle  this  problem  may  be 
reduced  to  a  problem  in  statics  as  follows  :  Given  a  stationary 
hoop  of  radius  r  on  each  unit  length  of  circumference  of  which 
an  outward  force  /(=  ^ir^n^rm')  acts,  required  the  tension  F  of 
the  hoop. 

By  applying  the  principle  of  virtual  work  the  relation  between 
the  tension  F  and  the  outward  force  /  per  unit  length  of  cir- 
cumference may  be  determined  as  follows.     Imagine  the  bolt   b 


122  ELEMENTS   OF   MECHANICS. 

to  be  shortened  by  a  certain  small  amount  /,  all  forces  remaining 
unchanged  in  value  ;  the  work  done  in  thus  shortening  the  bolt 
would  be  Fl ;  but  to  shorten  the  bolt  by  the  amount  /  would 
shorten  the  radius  of  the  hoop  by  //27r,  so  that  each  unit  of 
the  circumference  would  move  inwards  through  the  distance 
//27r  against  the  force  /,  and,  since  there  are  27rr  units  of 
circumference,  or  27rr  forces  like  /,  the  total  work  done  by  the 
contracting  hoop  would  be  27rrx/x//27r  which  is  equal  to 
rfl.  The  work  which  would  be  done  in  shortening  the  bolt,  how- 
ever, is  the  work  that  would  be  expended  in  contracting  the 
hoop,  so  that   Fl  =  rfl  or 

F=^rf 

in  which  F\?>  the  tension  of  a  hoop  of  radius  r,  and /is  the  out- 
ward force  pushing  on  each  unit  of  circumference  of  the  hoop.  If 
nothing  pushes  out  on  the  hoop,  however,  then  the  tension  F 
produces  an  unbalanced  inward  force  equal  to/,  which  unbal- 
anced force  suffices  to  constrain  each  particle  of  the  hoop  to  its 
circular  path,  so  that,  writing  ^ir^f^nn'  for /in  the  above  equa- 
tion, we  have 

F  =  AfTT^r^r^m' 

which  is  identical  to  equation  (i  i)  of  Art.  40. 

Problems. 

66.  A  165 -pound  man  climbs  a  height  of  40  feet  in  1 1  seconds. 
How  much  work  is  done,  and  at  what  rate  ?  Express  the  work 
in  foot-pounds,  and  in  joules;  and  express  the  power  in  foot- 
pounds per  second,  in  horse-power,  and  in  watts. 

67.  A  horse  pulls  upon  a  plow  with  a  force  of  100  pounds 
weight  and  travels  3  miles  per  hour.  What  power  is  developed  ? 
Express  the  result  in  foot-pounds  per  second,  in  watts,  and  in 
horse-power. 

68.  A  belt  traveling  at  a  velocity  of  70  feet  per  second  trans- 
mits 360  horse-power.  What  is  the  difference  in  the  tension  of 
the  belt  between  the  tight  and  loose  sides  in  pounds  weight  ? 


FRICTION.     WORK   AND    ENERGY.  1 23 

69.  A  stream  furnishes  500  cubic  feet  of  water  per  second  at  a 
head  of  1 5  feet.  What  power  can  be  developed  from  this  stream 
by  a  water  wheel  of  which  the  efficiency  is  60%? 

70.  The  engines  of  a  steamship  develop  20,000  horse-power, 
of  which  30  per  cent,  is  represented  in  the  forward  thrust  of  the 
screw  in  propelling  the  ship  at  a  speed  of  17  miles  per  hour. 
What  is  the  forward  thrust  of  the  screw  in  pounds-weight  ? 

71.  An  electric  motor  has  an  efficiency  of  80  per  cent,  and 
electrical  energy  costs  5  cents  per  kilowatt-hour.  How  much 
does  the  output  of  the  motor  cost  per  horse -power  hour  ? 

72.  A  1,000  horse-power  boiler  and  engine  plant  costs  about 
;^70,ooo  complete,  including  land,  building,  boilers,  engines  and 
auxiliary  apparatus  such  as  pumps  and  feed  water  heaters.  The 
cost  of  operating  this  plant  continuously,  night  and  day,  is  as 
follows : 

Interest  on  investment 5  P^'"  cent,  per  annum. 

Depreciation lo    **      **  **         ** 

Maintenance  and  repairs 4    <<      <<       <'         " 

Taxes  and  insurance  2    "      *'       **         '* 

Labor  ^30  per  day,  365  days  in  the  year. 

Coal  ;552.<X)  per  ton. 

The  average  demand  for  power  is  50  per  cent,  of  the  rated 
power  output  of  the  plant,  that  is  500  horse-power,  and  the  con- 
sumption of  coal  is  2j4  pounds  per  horse-power-hour.  Find  the 
cost  of  a  horse-power-hour  delivered  by  the  engine.  Ans. 
0.83  cent. 

73.  The  above  engine  will  drive  a  700  kilowatt  dynamo,  that  is 
a  dynamo  capable  of  delivering  700  kilowatts.  The  cost  of 
dynamo,  station  wiring  and  switch-board  apparatus  is  ;^20,ooo. 
The  average  output  of  the  dynamo  is  350  kilowatts  (correspond- 
ing to  500  horse-power  output  of  engine).  Calculate  the  cost  of 
electrical  energy  per  kilowatt-hour  at  the  station,  allowing  21  per 
cent,  for  interest,  depreciation,  etc.,  on  the  electrical  machinery 
and  allowing  $^  per  day  additional  for  labor.     Ans.  1.3 1  cents. 

74.  A  steam  engine  indicator  shows  an  average  steam  pressure 


124  ELEMENTS   OF   MECHANICS. 

of  55  pounds  per  square  inch  (reckoned  above  atmospheric  pres- 
sure) during  each  stroke  of  a  steam  engine,  and  the  engine  exhausts 
into  a  condenser  where  the  pressure  is  1 3  pounds  per  square  inch 
below  atmospheric  pressure.  The  diameter  of  the  piston  is  16 
inches,  the  diameter  of  the  piston  rod  is  3  inches,  the  length  of 
stroke  is  24  inches,  and  the  engine  makes  75  revolutions  per 
minute.     Find  the  indicated  horse-power  of  the  engine. 

75.  A  brake  test  of  a  steam  engine  gave  the  following  data : 
speed  of  engine  200  revolutions  per  minute,  length  of  brake  arm 
{a,  Fig.  49)  7  J  feet,  observed  force  at  end  of  brake  arm  and  at 
right  angles  to  arm  240  pounds-weight.  Find  the  brake  horse- 
power of  the  engine. 

76.  A  fan  blower  is  mounted  on  a  cradle  which  swings  on  knife 
edges  in  the  line  of  the  axis  of  the  fan.  When  the  fan  is  driven, 
the  cradle  tends  to  tip  to  one  side  and  this  tendency  is  balanced 
by  a  weight  sliding  on  a  horizontal  lever  arm,  as  shown  in  Fig. 
76/.    The  belt  is  thrown  off  the  fan,  the  sliding  weight  moved  to 


Fig.  76A 

give  a  balance  and  the  ''  zero  position  "  of  the  weight  is  observed. 
The  fan  is  then  driven  at  a  speed  of  1800  revolutions  per  minute 
and  the  weight  (10  pounds)  has  to  be  moved  6f  inches  from  its 
zero  position  to  balance  the  driving  torque  exerted  by  the  belt  on 
the  fan.  Find  the  power  expended  in  driving  the  fan  and  express 
it  in  horse -power  and  in  watts. 

77.  Four  idle  pulleys  A,  B,  C  and  D,  Fig.  y/p,  are  mounted 
in  a  frame  which  is  free  to  rotate  about  the  point  0  which  is  the 


FRICTION.     WORK   AND    ENERGY. 


125 


point  of  intersection  of  the  left  hand  stretches  of  belt.  A  weight 
W  slides  along  a  lever  arm  which  is  fixed  to  the  rocking  frame 
so  that  the  tilting  action  of  the  right  hand  stretches  of  belt  may 
be  balanced  and  measured.  When  no  power  is  transmitted  by 
the  belt,  the  tension  of  the  belt  is  the  same  everywhere  and,  under 
these  conditions,  the  weight  Wis  adjusted  to  its  zero  position  to 
give  a  balance.  When  power  is  transmitted  to  a  given  machine  the 
belt  tensions  F'  and  F^'  differ  and  the  weight  ^is  moved  away 


motion 


Ai       >  F^ 


Fig.  77A 


from  0  to  again  give  a  balance.  This  movement  of  the  weight  W 
from  its  zero  position  is  16  inches,  the  mass  of  ^is  55  pounds, 
the  distance  r  is  24  inches,  and  the  speed  of  the  belt  is  80  feet  per 
second.      Find  the  power  transmitted  by  the  belt. 

78.  A  shaft  transmits  1 00  horse-power  and  runs  at  a  speed  of 
250  revolutions  per  minute.  Calculate  the  torque  exerted  on  the 
shaft.  Express  the  result  in  pound-feet,  in  pound-inches,  and  in 
dyne-centimeters. 

79.  A  steamship  has  a  gross  mass  of  25,000  tons.  What  is 
the  kinetic  energy  of  the  ship  at  a  speed  of  1 8  miles  per  hour  ? 
Express  the  result  in  foot-pounds  and  in  horse-power  hours. 

80.  A  bicycle  rider  has  a  50-foot  hill  to  climb.  What  velocity 
must  he  have  at  starting  to  relieve  him  from  the  doing  of  one 
third  of  the  work  required  ? 

81.  The  rim  of  the  fly-wheel  of  a  metal  punch  is  5  feet  in 
diameter  and  its  mass  is  560  pounds.     At  what  initial  speed  must 


126 


ELEMENTS   OF   MECHANICS. 


the  fly-wheel  run  in  order  that  the  punch  may  exert  a  force  of 
72,000  pounds  through  a  distance  of  one  inch  and  reduce  the 
speed  of  the  fly-wheel  only  30  per  cent.  ? 

82.  A  counterpoise  of  ^  pound  balances  a  weight  of  100 
pounds  wherever  the  weight  may  be  placed  on  the  platform  of 
a  balance  scale.  In  what  way  and  to  what  extent  does  the  plat- 
form move  when  the  counterpoise  moves  ^  inch  downwards  ? 

83.  A  screw-jack  is  turned  by  a  lever  of  which  the  radial 
length  is  1 8  inches,  and  the  pitch  of  the  screw  is  ^  inch.  What 
is  the  lifting  force  produced  by  a  pull  of  100  pounds  on  the  end 
of  the  lever,  neglecting  friction  ? 

Note.  —  Consider  the  distance  travelled  by  the  end  of  the  lever  and  the  travel  of 
the  screw  in  one  complete  turn,  and  apply  the  principle  of  virtual  work. 

84.  The  differential  pulley  consists  of  a  large  pulley  A  and  a 
smaller  pulley  B  made  in  one  piece,  and  a  third  pulley  C  all 
threaded  with  an  endless  chain  as   shown  in  Fig.   84/.      The 


Fig.  84/. 


Fig.  85/. 


pulleys  A  and  B  are  sprocket  wheels  with  notches  which  en- 
gage the  links  of  the  chain  so  that  the  chain  cannot  slip  on  A 
and  B.  One  turn  of  A  and  B  takes  in  at  a  2.  length  of  chain 
which  is  equal  to  the  circumference  of  the  larger  pulley  A  and 
pays  out  at  ^  a  length  of  chain  which  is  equal  to  the  circum- 
ference of  the  smaller  pulley  B. 


FRICTION.     WORK   AND    ENERGY. 


27 


The  circumference  of  A  contains  1 2  notches,  the  circumference 
of  B  contains  1 1  notches  and  the  length  of  each  link  of  the  chain 
IS  lyi  inches.  What  is  the  lifting  force  produced  by  a  pull  of 
150  pounds  at  F,  neglecting  friction  ? 

85.  A  steel  hoop  i  o  feet  in  diameter  is  clamped  around  a  large 
wooden  tank  by  means  of  the  bolt  b,  Fig.  85/.  The  tension  in 
the  strap  is  1000  pounds- weight  (=  force  exerted  by  the  bolt), 
find  the  force  exerted  by  each  foot-length  of  hoop  against  the 
tank. 


-^v 


CHAPTER  VII. 
ROTATORY  MOTION. 

61.  Rotation  about  a  fixed  axis.  Definitions.  —  The  simplest 
case  of  rotatory  motion  is  that  which  is  exempHfied  by  the  rotation 
of  a  wheel  about  a  fixed  axis.  We  shall  first  consider  this  simple 
case  in  detail  and  then  proceed  to  the  more  complicated  rotation 
about  a  moving  axis.  In  order  to  rivet  the  attention  to  rotatory 
motion  to  the  exclusion  of  movements  of  distortion,  the  idea  of  a 
I'igid  body  will  be  used  throughout  the  chapter,  a  rigid  body  being 
an  ideal  body  which. cannot  change  its  shape  or  size. 

Angular  velocity.  —  Let  ^  be  the  angle  turned  by  a  rotating 

body  during  t  seconds  ;  the  quotient    c^//   is  called  the  average 

angular  velocity  of  the  body  during  the  t  seconds.     If  the  time 

interval  is  very  short,  the  quotient  A(/)/A/   is  the  actual  angular 

velocity  of  the  body  at  the  given  instant,    A</)    being  the  angle 

turned  by  the  rotating  body  during  the  short  time  interval    A/. 

When  the  angle  c/>  is  expressed  in  radians  and  time  t  in  seconds, 

then  the  quotient  (/>//   is  in  radians  per  second.     Angular  velocity 

is  expressed  in  radians  per  second  throughout  this  chapter.     In 

practice,  angular  velocity  is  generally  expressed  in  revolutions  per 

second.     There  are    27r    radians  in  one  revolution,  and  therefore 

one  revolution  per  second  is  equal  to    27r   radians  per  second,  or, 

in  general,  ^-r'r^ 

ft)  =  2irn  (28) 


in  which  ft)  is  the  angular  velocity  of  a  body  in  radians  per  second 
and  n  is  the  angular  velocity  in  revolutions  per  second. 

Angular  acceleration.  —  In  many  machines  a  part  may  rotate  at 
a  variable  angular  velocity.  This  is  most  strikingly  illustrated 
by  the  motion  of  the  balance  wheel  of  a  watch.  The  rate  of 
change  of  the  angular  velocity  of  a  body  is  called  its  angular 
acceleration.     Thus  an  engine  is  started,  and  after  six  seconds  the 

128 


ROTATORY   MOTION. 


129 


'fly  wheel  has  an  angular  velocity  of  4  revolutioms  per  second 
(=  25.13  radians  per  second),  so  that  the  average  angular  accel- 
eration of  the  wheel  during  the  six  seconds  is  4.1888  radians  per 
second  per  second.  Of  course,  the  fly  wheel  may  have  gained 
most  of  its  angular  velocity  during  a  portion  of  the  six  seconds, 
so  that  4.1888  radians  per  second  per  second  is  merely  its  aver- 
age angular  acceleration.  The  angular  acceleration  of  a  rotating 
body  at  a  given  instant  is  equal  to  the  quotient  Aco/At  where 
Ao)  is  the  angular  velocity  gained  during  the  short  interval  of 
time  A^. 

62.  Unbalanced  torque  and  angular  acceleration.  Definition  of 
moment  of  inertia.  —  When  a  wheel  is  set  in  rotation,  an  unbal- 
anced torque  must  act  upon  the  wheel.  This  is  exemplified  in 
the  operation  of  spinning  a  top.  When  a  rotating  wheel  is  left 
to  itself  it  loses  its  angular  velocity  and  comes  to  rest  on  account 
of  the  friction  of  the  wheel  against  the  air  and  on  account  of  the 


wheel 


Fig.  53a. 


Fig.  53/'. 


friction  of  the  shaft  in  its  bearings.  To  maintain  a  steady  motion 
of  rotation  of  a  wheel,  a  driving  torque  must  act  upon  the  wheel 
sufficient  to  balance  the  opposing  torque  due  to  friction. 
•  The  effect  of  an  unbalanced  torque  is  to  change  the  angular 
velocity  of  a  wheel,  or,  in  other  words,  to  produce  angular 
acceleration,  positive  or  negative  as  the  case  may  be.  The 
9 


I30  ELEMENTS   OF   MECHANICS. 

angular  acceleration  of  a  given  wheel  is  proportional  to  the  un- 
balanced torque  which  acts  upon  the  wheel,  and  a  given  unbal- 
anced torque  produces  a  small  angular  acceleration  of  a  large 
heavy  wheel,  or  a  large  angular  acceleration  of  a  small  light 
wheel.  Thus,  if  a  cord  be  wrapped  around  the  shaft  upon  which 
a  wheel  is  mounted,  a  pull  on  the  cord  produces  torque  equal  to= 
Fly  Figs.  53<a:  and  53^;  and  this  torque  imparts  angular  velocity 
to  the  small  light  wheel,  Fig.  53<3;,  at  a  rapid  rate,  whereas  it 
imparts  angular  velocity  to  the  large  heavy  wheel.  Fig.  53^,  at  a 
much  slower  rate. 

When  a  wheel  is  rotating  every  particle  of  the  wheel  moves  at 
a  definite  linear  velocity,  and  when  the  angular  velocity  of  the 
wheel  increases  it  is  evident  that  the  linear  velocity  of  every 
particle  of  the  wheel    must   increase ;    that   is   to   say,   angtilar 

acceleration  of  a  wheel  involves 
linear  acceleration  of  every  par- 
ticle in  the  wheel,  and  it  is  pos- 
sible to  show  the  exact  relation 
between  angular  acceleration  and 
the  unbalanced  torque  which  pro- 
duces it,  by  considering  the  linear 
acceleration  of  each  particle  in  a 
wheel.  The  following  discussion 
of  this  matter  is  the  foundation  of 
Pj  the  dynamics  of  rotatory  motion 

and    it    leads    to  a  definition  of 
what  is  called  the  7noment  of  inertia  of  a  wheel. 

Figure  54  represents  a  wheel  rotating  ;/  revolutions  per  second, 
or  2'Kn  radians  per  second,  about  the  axis  0.  The  particle  Aw 
describes  a  circular  path  of  which  the  circumference  is  27rr,  the 
particle  traces  this  circumference  7i  times  per  second,  and  therefore 
the  linear  velocity  v  of  the  particle  is  2'Krn  centimeters  per 
second,  r  being  expressed  in  centimeters  ;  but  2'Trn  is  equal  to 
the  angular  velocity  w  of  the  wheel  in  radians  per  second,  and, 

t^^^^f^^^  v=r<o  (29) 


ROTATORY   MOTION.  13V 

If  the  angular  velocity  of  the  wheel  is  changing,  the  linear 
velocity  v  of  the  particle  7n  must  change  r  times  as  fast  as  co, 
inasmuch  as  v  is  always  r  times  as  large  as  co.  Therefore, 
representing  the  angular  acceleration  of  the  wheel  by  a  (rate  of 
change  of  co)  and  representing  the  linear  acceleration  of  the 
particle  by  a  *  (rate  of  change  of  7/)  we  have 

a  =  ra  ,v  ^    "  (30) 

In  order  to  produce  the  acceleration  a  of  the  particle,  an  un- 
balanced force  F,  see  Fig.  54,  must  act  on  the  particle  in  the 
direction  of  a,  this  force,  expressed  in  dynamic  units,  must  be 
equal  to  A7n.a  according  to  equation  (3),  and  the  torque  action 
of  this  force  is  equal  to  Fr(=  Am. a  X  ^)  ;  but  a  is  equal  to  ra 
according  to  equation  (30),  so  that  Fr  =  Am.r^a,  or  representing 
Fr  by  ATj  we  have 

AT=  ar^'Am 
in  which    AT"  is  that  part  of  the  unbalanced  torque  7" acting  on 
the  wheel,   which    causes  the    linear  acceleration  of  the    given 
particle   Am.     Consider  in  this  way  all  of  the  particles  of  the 
wheel  and  we  have 

AT—  ar^'Am  . 

A7J  =  ar^'  Am^ 

AT^  —  ar^-Am^ 

etc.,  etc.,  whence,  by  adding,  we  have 

T=  a{f-  •  Am  +  r^^  •  Am^  +  r^  •  Am^  +  •  •  •) 
or 

T  =  a2,r--Am 
or  writing 

K^^^r-Am  (31) 

we  have 

T^Ka  ■        (32) 

^  We  are  not  concerned  here  with  the  radial  acceleration  of  the  particle  Aw,  since 
the  radial  acceleration  is  produced  by  unbalanced  radial  forces  which  have  no  torque 
action  about  O.  Radial  accelerations  of  the  particles  of  a  wheel  have  nothing  to  do 
with  the  angular  acceleration  of  the  wheel. 


ELEMENTS   OF   MECHANICS. 


The  quantity  A",  which  is  obtained  by  multiplying  the  mass  of  each 
particle  of  the  wheel  by  the  square  of  its  distance  from  the  axis 
and  adding  all  of  these  products  together,  is  called  the  moment  of 
inertia  of  the  wheel,  and  equation  (32)  shows  that  the  unbalanced 
torque  acting,  on  a  wheel  is  equal  to  the  product  of  the  moment 
of  inertia  of  the  wheel  and  the  angular  acceleration  of  the  wheel. 
Units  involved  in  equations  (31)  and  {^2).  If  c.g.s.  units  are 
used  throughout,  then  moment  of  inertia  is  expressed  in  grams 
X  centimeters  squared  (gr.  cm.^),  torque  is  expressed  in  dynes 
X  centimeters  and,  of  course,  angular  acceleration  is  expressed  in 
radians  per  second  per  second.  Equations  (31)  and  (32)  hold 
good,  however,  when  moment  of  inertia  is  expressed  in  pounds 
X  feet  squared  (lb.  ft.^),  torque  in  poundals  x  feet  and  angular 
acceleration  in  radians  per  second  per  second.  If  torque  is 
expressed  in  pounds-weight  x  feet,  moment  of  inertia  in  pound- 
feet^  and  angular  acceleration  in  radians  per  second  per  second, 
then  equation  (32)  becomes 

T^^i^Ka  (33) 

approximately. 

Example  of  the  calculation  of  moment  of  inertia.     The  moment  of  inertia  of  a 

homogeneous  soHd  of  regular  form  can  be  cal- 
culated by  the  methods  of  calculus.  Consider, 
for  example,  a  long  slim  rod  of  length  L  and 
mass  M  rotating  about  its  middle  point  O  as 
shown  in  Fig.  55.  The  mass  of  the  short  portion 
dr  is  AIj  L  y^dr  and  its  distance  from  O  is  r. 
Therefore,  writing  Af/ Ly^dr  for  Am  in  equa- 
tion (31 )  we  have 


but  tj^um  (integral)  Ir^dr  between  the  limits 
r^^K^l2  and  r  =  —  Z/2  is  equal  to  jV^^ 
soihm"  Jsr=^:rML^.  The  moments  of  inertia 
given  in  the  table  on  following  page  were  cal- 
culated in  this  way. 

Radius  of  gyration.  —  The  radius 
of  gyration  of  a  rotating  body  is  the 


Fig.  55. 


distance  p  from  the  axis  of  rotation  at  which  the  entire  mass  M 


ROTATORY   MOTION.  1 33 

TABLE. 
Moments  of  inertia  of  some  regular  homogeneous  solids. 


Axis  of  rotation  passing  through  center  of  mass. 


Sphere  of  radius  R  and  mass  M. 

Cylinder  of  radius  R  and  mass  J/,  axis  of  cylinder  is  the  axis 
rotation 

Slim  rod  of  length  L  and  mass  M^  axis  of  rotation  at  right  angles 
to  rod 

Rectangular  parallelopiped  of  length  L  and  breadth  By  axis  of  ro- 
tation at  right  angles  to  L  and  B 


Value  of  K. 


^\M{L^+B^) 


of  the  body  might  be  concentrated  without  altering  the  moment 
of  inertia  of  the  body.  If  the  entire  mass  M  were  concentrated 
at  distance  p  from  the  axis,  the  moment  of  inertia  would  be  equal 
to  Mpi^,  according  to  equation  (31).     That  is 


K^Mp'  (34) 

or 


'4. 


Using  the  values  of  K  in  the  above  table,  this  equation  shows 
that  the  radius  of  gyration  of  a  sphere  is  i/|  times  the  radius  of 
the  sphere,  the  radius  of  gyration  of  a  cylinder  rotating  about 
its  axis  of  figure  is  l/^^  times  the  radius  of  the  cylinder,  and  the 
radius  of  gyration  of  a  long,  slim  rod  rotating  about  an  axis  at 
right  angles  to  the  rod  and  passing  through  its  center  of  mass  is 
Vj2  times  the  length  of  the  rod. 

If  a  rotating  body  be  imagined  to  be  divided  into  particles  of 
equal  mass  then  the  radius  of  gyration  may  be  defined  as  the 
square-root-qf-the-average-square  of  the  distances  of  all  the  par- 
ticles from  the  axis. 

63.  Kinetic  energy  of  a  rotating  body.  — A  rotating  wheel  evi- 
dently stores  kinetic  energy  because  it  can  do  work  while  being 
brought  to  rest.  The  kinetic  energy  of  a  rotating  body  is  given 
by  the  equation 

W^iKco'  (35) 

in  which  everything  is  expressed  in  c.g.s.  units.     The  proof  of 


134  ELEMENTS   OF   MECHANICS. 

this  equation  gives,  perhaps,  a  clearer  idea  of  the  significance  of 
moment  of  inertia  than  the  discussion  of  equation  (32)  given  in 
the  foregoing  article.  Consider  the  rotating  wheel  shown  in  Fig. 
54.  The  linear  velocity  of  the  particle  t^m  is  ro),  according  to 
equation  (29),  and  therefore  the  kinetic  energy  of  this  particle  is 

according  to  equation  (26). 

Consider  in  this  way  all  of  the  particles  of  the  wheel  and  we 
have 

AH^  =  JA7;/-rW 

A^Fi  =  JA;;/^-r>2 

etc.         etc. 
whence,  by  adding,  we  have 

and  by  comparing  this  equation  with  equation  (31)  we  have 
equation  (35). 

64.  Relation  between  moments  of  inertia  about  parallel  axes.  —  Let  K  be 

the  moment  of  inertia  of  a  body  of  mass  M  about  a  given  axis  passing  through  the 
center  of  mass  of  the  body,  and  let  K^  be  the  moment  of  inertia  of  the  body  about 
another  axis  parallel  to  the  first  and  distant  d  from  the  center  of  mass,  then 

K^=KArd^M  (36) 

Let  O,  Fig.  56,  be  the  center  of  mass  of  the  body,  chosen  as  the  origin  of  co- 
ordinates, let  K  be  the  moment  of  inertia  of  the  body  about  an  axis  through  O  per- 
pendicular to  the  plane  of  the  paper,  and  let  K'  be  the  moment  of  inertia  of  the  body 
about  an  axis  through  O'  also  perpendicular  to  the  plane  of  the  paper.  Consider  a 
sample  particle  of  the  body  Aw,  distant  r  from  O  and  distant  r^  from  O^^  and  of 
which  the  coordinates  are  x  and  y. 

By  trigonometry  we  have 

r^^  =r'^J^d'^—  ira  cos  ^  (i) 

^\      From  equation  (31)  we  have 

K^  =  l.r^'^Lm  (ii) 


ROTATORY   MOTION.  1 35 

whence,  substituting  the  value  of  r'  from  equation  (i)  we  have 

K'  =  2r2  .  Am  +  Id'^Am  —  2d^r  cos  d  •  Am  (iii) 

but  2r2  •  Am  is^  equal  to  A',  and  l^d^Am  is  equal  to  d'^M.     Furthennore  l^r  cos  Q  •  Am 


Fig.  56. 

is  equal  to  ^x  •  Az;?,  which  is  equal  to  zero  according  to  equation  (22)  since  the  origin 
of  coordinates  is  chosen  at  the  center  of  mass  of  the  body.  Therefore  equation  (iii) 
reduces  to  equation  (36) 

65.  Equivalent  mass  of  a  rolling  wheel.  —  To  set  a  rolling 
wheel  in  motion  it  is  necessary  not  only  to  impart  linear  velocity 
to  the  wheel  but  also  to  set  the  wheel  rotating,  so  that  a  given 
unbalanced  force  produces  less  accelera|^n  than  it  would  produce 
if  the  wheel  were  lifted  from  its  track  and  allowed  to  move  with- 
out rotation.  Insofar  as  the  relation  between  force  (unbalanced) 
and  acceleration  is  concerned,  a  rolling  wheel  may  be  treated  as  a 
body  performing  simple  translatory  motion  by  assigning  to  the 
wheel  a  mass  in  excess  of  its  actual  mass.  This  fictitious  mass 
of  a  rolling  wheel  is  called  its  equivalent  mass,  and  it  may  be 
defined  as  that  mass  M  which  would  store  the  same  amount  of 
kinetic  energy  as  the  rolling  wheel  at  a  velocity  equal  to  the 
linear  velocity  v  of  the  wheel.     Thus  we  may  write 


IMv^  =  Imv"  +  IKco'' 


(0 


in  which  m  is  the  actual  mass  of  the  wheel,  K  is  the  moment  of 
inertia  of  the  wheel,  and  co  is  the  angular  velocity  of  the  wheel. 


136 


ELEMENTS   OF   MECHANICS. 


The  angular  velocity  o),  however,  satisfies  the  equation 

V  —  r(a  (29)  bis 

and  therefore,  substituting  vjr  for  «  in  equation  (i)  we  have 

K 


or 


K 


M=m  + 


^ 


(37) 


Examples.  —  {d)  The  rolling  motion  of  the  wheels  of  a  railway 
train  causes  the  train  to  behave,  insofar  as  acceleration  and  kinetic 
energy  relations  are  concerned,  as  if  its  mass  were  greater  by  the 
amount  nK\7^  than  its  actual  mass,  where  n  is  the  number  of 
wheels,  r  is  the  diameter  of  the  rolling  circle  of  each  wheel,  and 
K  is  the  moment  of  inertia  of  each  wheel.  In  the  case  of  an  elec- 
tric car  with  a  geared  motor,  the  moment  of  inertia  of  the  motor 


mgSiad 


Fig.  57. 

armature  can  be  reduced  to  an  equivalent  moment  of  inertia  of 
wheel  and  thus  be  included  in  the  value  of  K  in  equation  (37),  by 
multiplying  the  moment  of  inertia  of  the  motor  armature  by  the 
square  of  the  gear  ratio  (ratio  of  the  diameters  of  the  rolling 
circles  of  the  two  gears). 

(d)  Consider  a  metal  sphere  of  mass  m  and  radius  r  rolling 
down  an  inclined  plane  as  shown  in  Fig.  57.  The  vertical  pull 
of  the  earth  mg-  has  a  component  parallel  to  the  plane  which  is 
equal  to    mg-  sin  6,    and  this  force  would  cause  the  ball  to  move 


ROTATORY   MOTION. 


^?>7 


with  an  acceleration  equal  to  g  sin  6  if  it  were  not  for  the  rota- 
tory motion ;  but  on  account  of  the  rotatory  motion  the  sphere 
behaves  as  li  its  mass  were  |^  times  m,  according  to  equation  (37), 
and  therefore  it  rolls  down  the  plane  with  an  acceleration  of  only 
I  of  ^sin  6.  Friction  is,  of  course,  neglected. 
^  A  wheel  and  axle  rolling  on  a  track  as  shown  in  Fig.  58,  has  a 


Fig.  58. 


rolHng  circle  of  small  radius  r,  so  that  its  equivalent  mass  is  very 
large,  according  to  equation  (37),  and,  therefore,  such  a  wheel  and 
axle  rolls  down  an  inclined  plane  with  a  very  small  acceleration. 
66.  Correspondence  between  tiranslatory  motion  and  rotatory 
motion.  To  every  equation  in  translatory  motion  there  is  a 
corresponding  equation  in  rotatory  motion  in  which  moment  of 
inertia  K  takes  the  place  of  mass  m,  angle  takes  the  place  of 
distance,. angular  velocity  &>  takes  the  place  of  hnear  velocity  Vy 
angular  acceleration  a  takes  the  place  of  linear  acceleration  a^ 
and  so  on.  The  following  table  exhibits  the  pairs  of  correspond- 
ing equations. 

TABLE 


Translatory  motion 

Rotatory  motion 

F=ma                    (3) 

T=JCa                (32) 

VV^Fd                (24) 

W=  T<1>                    (i) 

P=Fv                 (25) 

P=  7w                  (ii) 

W-=  %mv^           (26) 

W=%Ko.^         (35) 

F=-kx             (15) 

^=-^0            (38) 

k^^TT^nhn          (16) 

K^^TTht^K        (39) 

Of  these  equations,  those  numbered  (i),  (ii),  (38)  and  (39)  have 
not  been  previously  discussed  ;  equation  (i)  refers  to  the  work 


13^  ELEMENTS   OF   MECHANICS. 

W  done  by  the  torque  T  in  turning  a  body  through  an  angle  </>, 
axis  of  torque  and  axis  of  motion  being  coincident ;  and  equation 
(ii)  refers  to  the  power  P  developed  by  a  torque  T  which  acts  on 
a  body  rotating  at  angular  velocity  o),  axis  of  torque  and  axis  of 
motion  being  coincident. 

Equations  (38)  and  (39)  refer  to  harmonic  rotatory  motion, 
that  is,  to  oscillatory  motion  about  an  axis,  such  as  is  exemplified 
by  the  motion  of  the  balance  wheel  of  a  watch. 

The  equations  of  circular  translatory  motion  correspond  to  the 
equations  of  the  gyroscope  to  a  limited  extent  as  explained  in 
Art.  72. 

67.  Rotatory  harmonic  motion.  —  Consider  a  weight  suspended 
by  a  steel  wire.  The  weight  will  stand  in  equilibrium  with  the 
wire  untwisted.  If  the  weight  is  turned  around  the  wire  as  an 
axis  through  the  angle  (f)  from  this  equilibrium  position,  then  the 
twisted  wire  will  exert  a  torque  T  on  the  weight  tending  to  turn 
it  back,  and  this  torque  will  be  proportional  to  </>,  that  is 

T-=-K<j>  (38) 

in  which  the  factor  /c  is  a  constant  for  a  given  wire ;  it  is  called 

the  consta7it  of  torsion  of  the  wire. 

By  analogy  with  harmonic  translatory  motion  as  discussed  in 

Art.  42,  it  is  evident  from  equation  (38)  that  the  suspended  weight, 

if  started,  will  oscillate  about  the  wire  as  an  axis,  and  that  the 

number  n  of  complete  oscillations  per  second  will   satisfy  the 

equation 

K  =  47rVir  (39) 

or,  using  i  /t  for  «,  where  t  is  the  period  of  one  oscillation,  we 
have 

ic  =  -^^  (40) 

A  weight  hung  by  a  wire  and  set  oscillating  about  the  wire  as 
an  axis,  is  called  a  torsion  pendulum.  « 

68.  Use  of  the  torsion  pendulum  for  the  comparison  of  moments 
of  inertia. — The  constant  of  torsion,  /c,  equations  (39)  and  (40), 


ROTATORY   MOTION. 


139 


is  nearly  independent  of  the  amount  of  weight  supported '  by  the 
wire,  unless  the  weight  becomes  excessive,  therefore  if  two  bodies 
are  hung  from  the  same  wire,  one  at  a  time,  and  their  respective 
periods  of  torsional  vibration  t  and  t'  observed,  then  from  equa- 
tion (40)  we  have 

(0 

and 


K  = 


whence 


(2) 


(iii) 


from  which  K'  may  be  calculated  if  K  is  known.     For  example, 
one  of  the  suspended  bodies  may  be  a 
homogeneous  circular  disk  of  which  the 
moment  of  inertia  is  known  (see  table  in 
Art.  62). 

69.  The  gravity  of  pendulum  consists 
of  a  rigid  body  AB,  Fig.  59,  suspended 
so  as  to  be  free  to  turn  about  a  hori- 
zontal axis  (9.  Let  C,  Fig.  59,  be  the 
center  of  mass  of  the  body.  This  point 
C  is  vertically  below  O  when  the  body  is 
in  equilibrium.  Let  the  body  be  swung 
to  one  side  through  the  angle  (^,  as 
shown.  Then  the  force,  Mg^  with  which 
the  earth  pulls  the  body  tends  to  swing 
the  body  back  to  the  vertical  position 
with  a  torque  T  which  is  equal  to  the 
product  of   Mg  and  the  length  of  the  Fig.  59. 

arm  aC .      But  the  distance  aC  is  equal 
to  .rsin  <^,  where  ;ir  is  the  distance  OC.     Therefore 


7"=  Mgx^iVi  (j) 


(0 


I40  ELEMENTS    OF   MECHANICS. 

When  </>  is  small,  then  sin  </>  =  (/>,  very  nearly,  and  equation  (i) 
becomes 

r=yl/^^-<^*  (ii) 

Comparing  this  with  equations  (38)  and  (40),  we  find  that 

'^^--M,.  (41) 

in  which  K  is  the  moment  of  inertia  of  the  body  about  the  axis 
Oy  T  is  the  period  of  one  complete  pendulous  vibration  of  the 
the  body,  and  g  is  the  acceleration  of  gravity. 

A  pendulum  such  as  here  described  is  sometimes  called  a 
physical  pendulum  to  avoid  confusion  with  the  ideal  simple  pendu- 
lum described  in  Art.  43. 

The  simple  pendulum.  An  ideal  pendulum  consisting  of  a 
particle  of  mass  M  suspended  by  a  weightless  cord,  or  rod,  of 
length  /  is  called  a  simple  pendulum.  The  moment  of  inertia  of 
such  a  pendulum  about  the  supporting  axis  0  is  K=Ml^j  ac- 
cording to  equation  (31).  Furthermore,  the  center  of  mass  of  a 
simple  pendulum  is,  of  course,  at  the  center  of  the  suspended 
particle.  Therefore,  for  the  simple  pendulum,  we  may  write 
MP   for  K,  and  /  for  x  in  equation  (41),  whence  we  have 

V  ==^  (42) 

or 

^-p  (43) 

in  which  /  is  the  length  of  a  simple  pendulum,  t  is  the  period  of 
one  complete  vibration  of  the  pendulum  and  g  is  the  acceleration 
of  gravity. 

Equivalent  length  of  a  physical  pendulum.  The  length  of  a 
simple  pendulum  which  would  have  the  same  period  of  vibration 
as  a  given  physical  pendulum  is  called  the  equivalent  length  of 
the  given  physical  pendulum.     Now,  according  to  equation  (43), 

*  Of  course  this  equation  should  be  written  7"= — Mgx  -^  because  T  tends  to 
reduce  0. 


ROTATORY   MOTION.  I4I 

the  length  of  a  simple  pendulum,  of  which  the  period  of  one 
vibration  would  be,  r  is  /  =T^^/47r^.  Therefore,  solving  equation 
(41)  for  T^gj/^it^  (=0  ^^  have 

in  which  /  is  the  equivalent  length  of  a  given  physical  pendulum, 
K  is  the  moment  of  inertia  of  the  pendulum  about  its  axis  of 
support,  M  is  the  mass  of  the  pendulum,  and  x  is  the  distance 
from  the  point  of  support  to  the  center  of  the  mass  of  the 
pendulum. 

The  point  in  the  line  OC,  Fig.  59,  which  is  at  a  distance 
/(=  KlMx)  from  O  is  called  the  center  of  oscillation  of  the 
pendulum.  This  point  is  also  called  the  center  of  percussion  of 
the  pendulum  for  the  reason  that  if  the  pendulum  is 
started  or  stopped  by  a  horizontal  hammer  blow  at  this  [J* 
point  no  side  force  is  produced  on  the  supporting  axis. 
See  Art.  71. 

70.  The  determination  of  gravity.  —  The  most  accur- 
ate determination  of  the  acceleration  of  gravity  is  made 
by  means  of  the  pendulum.  This  determination  would 
be  a  very  simple  thing  if  it  were  feasible  to  construct  a- 
simple  pendulum,  in  which  case  equation  (42)  could  be 
used  for  calculating  gravity  from  the  measured  length,  /, 
of  the  simple  pendulum  and  its  observed  period  r.  The 
determination  of  the  acceleration  of  gravity  by  means  of 
an  actual  pendulum  depends,  however,  upon  the  deter- 
mination of  the  moment  of  inertia  of  the  pendulum,  as  is 
evident  from  equation  (41),  and  the  moment  of  inertia  of 
a  body  cannot  be  determined  with  great  accuracy.  This 
difficulty  is  obviated  by  means  of  the  so-called  reversion 

•^  ^  Fig.  60. 

pendulum  which  was  devised  by  Henry  Kater  in   18 18. 

A  simple  form  of  Kater' s  pendulum  is  shown  in  Fig.  60.  A 
stiff  metal  bar  has  two  knife-edges,  from  either  of  which  it  may 
be  swung  as  a  pendulum,  and  two  weights,  WW,  which  may 
be  adjusted  until  the  period  r  of  one  vibration  of  the  pendu- 


142  ELEMENTS    OF   MECHANICS. 

lum  is  the  same  whether  it  be  swung  from  a  or  b.  Then  the 
distance  between  the  knife-edges  a  and  b  is  the  equivalent  length 
of  the  pendulum  and  may  be  used  for  /  in  equation  (43). 

Comparison  of  the  values  of  gravity  at  two  places  by  means  of 
the  pendulum.  —  If  the  same  pendulum  be  swung  at  two  places  in 
succession  and  its  respective  periods  t  and  t'  observed,  we  have 
from  equation  (41) 

/i^K      __  ... 

''—^^Mgx  (1) 

and 

\n---Mg^x  (11) 

in  which  g  and  g'  are  the  respective  values  of  the  acceleration 
of  gravity  at  the  two  places.  Dividing  equation  (i)  by  equation 
(ii),  member  by  member,  we  have 

g'~  -^ 

From  this  equation  the  value  of  g  may  be  accurately  deter- 
mined at  any  place  in  terms  of  its  known  value  at  another  place, 
by  observing  the  values  of  r  and  t'  of  an  ordinary  pendulum, 
every  precaution  being  taken  to  avoid  variations  of  dimensions 
of  the  pendulum  due  to  temperature  or  to  careless  handling. 
Most  of  the  gravity  determinations  of  the  United  States  Coast 
and  Geodetic  Survey  are  made  in  this  way,  the  value  of  g  at 
Washington  having  been  once  for  all  determined  with  the  greatest 
possible  accuracy  by  means  of  Kater's  pendulum. 

The  opposite  table  gives  the  value  of  g  in  centimeters  per 
second  per  second  at  several  places  as  determined  by  the  pen- 
dulum. 

Theory  of  the  reversion  pendulum.  —  Consider  a  body  of  mass  J/,  its  center  of 
mass  at  (9,  Fig.  61.  Let  O^^  O,  and  O^^  be  co-linear  points  ;  let  r^  and  t^^  be  the 
vibration  periods  of  the  body  swung  as  a  pendulum  from  O^  and  O^^  respectively  ; 
and  let  A',  K'^  and  K^^  be  the  moments  of  inertia  of  the  body  about  6>,  O' ^  and  O^' 
respectively.     From  equation  (41)  we  have 


ROTATORY   MOTION. 
TABLE. 


143 


Locality. 


Boston,  Mass 

Philadelphia,    Pa.., 
Washington,  D.  C 

Ithaca,  N.  Y 

Cleveland,   O  

Cincinnati,  O 

Terre  Haute,  Ind. 

Chicago,   111 

St.   Louis,   Mo 

Kansas  City,  Mo.., 

Denver,  Col 

San  Francisco,  Cal 

Greenwich 

Paris 

Berlin 

Vienna 

Rome  

Hammerfest 


Latitude. 

42° 

21^ 

33^-^ 

39 

57 

06 

3« 

53 

20 

42 

27 

04 

41 

30 

22 

39 

08 

20 

39 

28 

42. 

41 

47 

25 

3« 

3« 

03 

39 

05 

50 

39 

40 

36 

37 

47 

00 

.■Ji 

29 

CX) 

48 

50 

II 

52 

30 

16 

48 

12 

35 

41 

53 

53 

70 

40 

00 

Longitude. 

71° 

03/ 

5c// 

75 

II 

40 

77 

01 

32 

76 

29 

00 

81 

36 

38 

84 

25 

20 

«7 

23 

49 

87 

36 

03 

90 

12 

13 

94 

35 

21 

04 

56 

55 

22 

26 

00 

0 

00 

00 

2 

20 

15 

13 

23 

44 

16 

22 

55 

12 

28 

45 

22 

38 

00 

Elevation. 


22  meters. 
16       " 


10 

247 
210 
245 
151 

182 

154 

278 

1638 

114 

47 
72 
35 
150 
53 


Value  of  ^  (not 

Reduced 

to  Sea-level). 


980.382 
980.182 
980.100 
980.286 
980.227 
979.990 
980.058 
980.264 
979.987 

979-976 
979-595 
979-951 

981.170 
980.960 
981.240 
980.852 
980.312 
982.580 


and 


From  equation  (36)  we  have 

K'  =  K^  x^M  (iii) 

K''=K-^y'^M  (iv) 

Substituting  these  values  of  K^  and  K'^  i"  (i)  and  (ii),  we  have 

47r2(^+  x'^M) 


=  Mgx 


Mgy 


Eliminating  A'/ M  from  (v)  and  (vi),  we  have 

4;r2(x2-y) 


XT^^  —  jr^/- 


(V) 

(vi) 
(45) 


(ii) 


If  T^=:T^^,  we  may  cancel  {x — y),  if  (jt: — y")    is  not  equal 
to  zerOy  giving 

^^^^^^,  .(46) 


o'1 


Fig.  61, 


(l)  If  the  pendulum  has  been  adjusted  by  repeated  trial,  so  that    r^:=r''',    then 


144  ELEMENTS    OF   MECHANICS. 

equation  (46)  enables  the  calculation  of  ^,  when  (x -\- ^)  and  t^  have  been  ob- 
served. 

(2)  If  the  pendulum  has  not  been  adjusted,  equation  (45)  enables  the  calculation 
of  g;  when  x,  y,  r^,  and  r'''  have  been  observed. 

(3)  If  the  pendulum  has  been  roughly  adjusted,  so  that  r-'  and  r^^  are  nearly 
equal,  then  equal  and  opposite  errors  in  x  and  y  very  nearly  annul  each  other  in 
their  influence  upon  the  value  of  g  as  calculated  by  equation  (45).  Therefore  equa- 
tion (45)  gives  g  very  accurately  when  r^  and  t"  are  nearly  equal,  (j;  -\- y^  being 
measured  with  great  accuracy,  and  x  measured  roughly.  The  value  of  y  is  taken 
from  {^x '\- y) — x,  so  that  its  error  may  counteract  the  error  due  to  the  roughly 
measured  value  of  x.  The  position  of  the  center  of  mass  C,  Fig.  61,  is  found  with 
sufficient  accuracy  for  the  rough  measurement  of  x  by  balancing  the  pendulum  hori- 
zontally on  a  knife  edge. 

Note. — When  x:=yy  equation  (45)  becomes  indeterminate,  giving  0/0=^,  and 
in  this  case  equation  (46)  is  not  necessarily  true,  since  it  has  been  derived  from  equa- 
tion (45)  by  cancelling    {^x — j),    which  is  zero. 

71.  Motion  of  a  rigid  body  when  struck  with  a  hammer. — 

When  an  unbalanced  force  continues  to  act  upon  a  body  for  an 
appreciable  length  of  time,  the  problem  of  determining  the  motion 
of  the  body  is  complicated  by  the  fact  that,  as  the  body  moves, 
the  force  generally  changes  its  point  of  application,  or  its  value, 
or  its  direction,  or  all  three  of  these  things  may  change  simultan- 
eously. The  force  due  to  a  hammer  blow,  however,  is  of  such 
short  duration  that  the  actual  movement  of  the  body  during  the 
time  that  the  force  acts  is  negligible  and  the  problem  of  finding 
the  motion  produced  by  the  hammer  blow  is  quite  simple.  A 
hammer  blow  is  called  an  impulse'^  and  it  is  measured  by  the 
product  of  the  average  value,  F,  of  the  force  exerted  by  the 
hammer  and  the  short  time  /  that  the  force  continues  to  act ;  that 
is,  an  impulse  is  expressed  in  terms  of  force  multipHed  by  time, 
in  dyne-seconds,  if  c.g.s.  units  are  employed. 

A  rigid  stick,  AB,  Fig.  62a,  is  struck  with  a  hammer  in  the 
direction  of  the  arrow,  h,  at  a  point  distant  x  above  the  center 
of  mass,  0,  of  the  stick.  The  motion  imparted  to  the  stick  by 
the  blow  is  a  combination  of  translatory  motion  and  rotatory 
motion,  but  the  combination  of  a  constant  translatory  motion 
and  a  constant  rotatory  motion  is  exactly  the  kind  of  motion 

*  The  impulse  of  ahammer  blow,  when  the  hammer  is  brought  to  rest  by  the  blow, 
is  equal  to  the  momentum  of  the  hammer.     See  Art.  47. 


ROTATORY   MOTION. 


45 


which  is  performed  by  a  rolling  wheel,  and  therefore  the  hammer 
blow  causes  the  stick  to  move  as  if  the  stick  were  attached  to  a 
circular  hoop,  CC^  and  this  hoop  allowed  to  roll  on  a  straight 
rail.  The  center  of  the  rolling  circle,  CC,  is  at  the  center  of 
mass  of  the  stick,  and  the  radius,  y,  of  the  rolling  circle  depends 
upon  the  distance,  x,  and  upon  the  ratio  of  the  moment  of  inertia 
of  the  stick  (about  O)  to  the  mass  of  the  stick,  according  to 
equation  (47).  At  the  instant  of  the  hammer  blow  the  motion  of 
the  stick  is  equivalent  to  a  simple  motion  of  rotation  about  the 
point  0'\ 

To  analyze  the  effect  of  the  hammer  blow,  the  translatory  mo- 


\. 


i|(^(li_|!|_^.  rolling  circle] 


9 


./ 


rail 


Fig.  62«. 


tion  and  the  rotatory  motion  may  be  treated  separately.  Re- 
garding the  translatory  motion,  we  know,  from  Art.  48,  that  the 
velocity  imparted  to  the  center  of  mass  is  the  same  as  if  the  whole 
mass  of  the  body  were  concentrated  there  and  acted  upon  directly 
by  the  total  force  of  the  hammer.  Let  F  be  the  average  force  due 
to  the  hammer,  and  /  the  time  (very  short)  that  it  continues  to  act. 
Then  F/M  is  the  acceleration  of  the  center  of  mass,  and  F/M 
multiplied  by  t  is  the  velocity  imparted  to  the  center  of  mass. 
As  to  the  rotatory  motion  of  the  body,  it  is  evident  that    Fx  is 


10 


146 


ELEMENTS   OF   MECHANICS. 


the  torque  about  0  due  to  the  force  of  the  hammer,  so  that  FxjK 
is  the  angular  acceleration  of  the  body  during  the  time  t,  and 
FxjK  multipHed  by  t  is  the  angular  velocity  imparted  to  the 
body  by  the  hammer  blow. 

Now  the  whole  body  is  moving  to  the  right  at  a  velocity  FtjM 
on  account  of  the  translatory  motion,  and  any  point  at  a  distance 
r  below  0  is  moving  to  the  left  at  a  linear  velocity  equal  to  r 
times  the  angular  velocity,  FtxJK,  therefore,  for  the  point  0"^ 
which  is  for  the  moment  stationary,  we  must  have,  writing  y  for  r, 


or 


Ftxy  _  Ft 
K 


(47) 


which  determines  the  radius  y  of  the  rolling  circle  when  x  and 

KjM  diVe  given. 

The  problem  of  the  base-ball  bat.  —  At  the  instant  that  a  base-ball 
bat  strikes  a  ball,  the  motion  of  the  bat  is  a 
simple  motion  of  rotation  about  a  certain 
point  0"  Fig.  ^2b ;  and,  if  the  distances 
X  and  y  satisfy  equation  (47),  then  the 
effect  of  the  impact  of  bat  and  ball  is  to 
reduce  the  angular  velocity  of  the  bat 
about  the  point  0"  without  moving  the 
point  0" .  The  point  of  a  bat  which  must 
strike  a  ball  so  that  the  impact  may  have 
no  tendency  to  change  the  position  of  the 
point  about  which  the  bat  is  rotating  at 
the  instant  of  impact,  is  called  the  center 
of  percussion  of  the  bat.  The  position 
of  the  center  of  percussion  depends  of 
course  upon  the  position  of  the  point  0" 
about  which   the  bat  is   rotating   at  the 

instant  of  impact. 


ROTATORY   MOTION. 


147 


72.  Precessional  rotatory  motion.  The  foregoing  articles  refer 
to  rotation  about  a  fixed  axis,  or,  as  in  the  case  of  a  rolling 
wheel,  to  rotation  about  an  axis  which  performs  translatory 
motion.  The  axis  of  a  rotating  body  may,  however,  change  its 
direction  continuously.  We  shall  discuss  here  only  the  com- 
paratively simple  case  *  in  which  a  symmetrical  body  spins  about 
its  axis  of  symmetry  while  at  the  same  time  the  axis  of  spin 
rotates  uniformly.  This  rotation  of  the  axis  of  spin  is  called 
precession^  and  the  axis  about  which  the  axis  of  spin  rotates  is 
called  the  axis  of  precession. 

The  gyroscope  consists  of  a  heavy  wheel  mounted  on  an  axle 
which  is  pivoted  in  a  metal  supporting  ring,  as  shown  in  Fig.  63. 
The  wheel  is  set  in  rapid 
rotation  by  wrapping  a 
cord  on  the  axle  and  giv- 
ing the  cord  a  vigorous 
pull.  When  the  wheel  is 
thus  set  rotating,  the  direc- 
tion of  the  axle  remains 
unaltered  as  long  as  no 
external  twisting  force,  or 
torque,  acts  upon  it ;  an 

unbalanced  torque  is  necessary  to  change  the  direction  of  the  axis 
of  a  rotating  body,  just  as  an  unbalanced  force  is  required  to  change 
the  direction  of  translatoiy  motion  of  a  particle. 

In  order  to  describe  precisely  how  an  unbalanced  torque 
changes  the  direction  of  the  axis  of  a  rotating  body,  it  is  very 
convenient  to  represent  angular  velocity  and  torque  by  lines  in  a 
diagram.  To  represent  an  angular  velocity  by  a  line,  draw  the 
line  in  the  direction  of  the  axis  of  spin  and  of  such  length  as  to 
represent  to  scale  the  value  of  the  angular  velocity  in  radians 
per  second  ;  to  represent  a  torque  by  a  line,  draw,  the  line  in  the 

*The  student  is  referred  to  Poinsot's  Theorie  Nouvelle  de  la  Rotation  des  Corps, 
which  is  perhaps  the  most  intelligible  account  of  the  motion  of  a  non-symmetrical 
rigid  body. 


^^r> 


Fig.  63. 


148 


ELEMENTS   OF   MECHANICS. 


direction  of  the  axis  of  the  torque  and  of  such  length  as  to 
represent  to  scale  the  value  of  the  torque  in  dyne-centimeters. 
In  each  case  an  arrow-head  is  to  be  placed  on  that  end  of  the 
line  towards  which  a  right-handed  screw  would  travel  if  turned 
in  the  direction  of  rotation  in  the  one  case,  or  if  turned  in  the 
direction  of  the  torque  in  the  other  case. 

Figure  64a  is  a  top  view  of  the  gyroscope  ;  the  metal  ring  rests 
upon  a  supporting  pivot  underneath  the  ring  at  0,  and  the  line 
OPy    Fig.  64^,  represents  the  angular  velocity  co  of  the  spinning 


<w"^w^^^: 


Fig.  64«. 


Fig.  643. 


wheel  at  a  given  instant.  The  pull  of  the  earth  on  the^wheel  and 
ring  produces  an  unbalanced  torque  about  the  axis  OT,  Fig. 
64a.  The  effect  of  this  unbalanced  torque,  during  a  short  in- 
terval of  time,  is  to  impart  to  the  wheel  an  additional  angular 
velocity  Ao)  about  the  axis  OT,  and  the  resultant  angular 
velocity  is  then  about  the  axis  OP^,  Fig.  64^  ;  that  is,  tke  effect 
of  the  unbalanced  torque  T  is  to  cause  the  axis  of  spin  to  sweep 
around  0  in  the  direction  of  the  arrozv  O. 

This  effect  of  an  unbalanced  torque  upon  a  rapidly  rotating 
body  is  also  exemplified  by  the  motion  of  a  spinning  top.     Thus 


ROTATORY    MOTION. 


149 


the  line  OP^  Fig.  65,  represents  the  angular  velocity  of  a 
spinning  top.  The  vertical  pull  of  the  earth,  mg,  produces  an 
unbalanced  torque  about  0^  and  the  angular  velocity  produced 
by  this  unbalanced  torque,  by  being  added  continuously  to  OP 
as  a  vector,  causes  the  axis  of  spin  OP  to  sweep  round  the  vertical 
axis    OV  in  the  direction  indicated  by  the  arrow  12. 

The  force  required  to  constrain  a  particle  to  a  circular  orbit 
depends  upon  the  mass  of  the  particle  and  upon  the  linear  accel- 
eration which  is  involved  in  the  continual  change  of  direction  of 
the  velocity  of  the  particle.     The  torque  required  to  produce  pre- 


O 

Fig.  65. 


n 


0) 


Fig.  66. 


cession  of  a  spinning  body  depends  upon  the  moment  of  inertia 
of  the  body  and  upon  the  angular  acceleration  which  is  involved 
in  the  continual  change  of  direction  of  the  axis  of  spin.     Preces- 
sional  motion  of  a  spinning  body  corresponds  to  circular  trans-_ 
jatory  motion. 

The  torque  required  to  produce  precessional  rotatory  motion 
may  be  derived*  from  equation  (32)  as  follows:  Let  the  line 

*  This  derivation  is  correct  only  when  the  rotating  body  is  symmetrical  with  respect 
to  the  axis  of  spin  and  also  to  the  axis  of  precession,  as  in  Figs.  64  and  69  ;  and  it  is 
approximately  correct  in  a  case  like  Fig.  65,  provided  the  angular  velocity  of  spin  is 
very  much  greater  than  the  angular  velocity  of  precession.  A  precessional  motion  of 
the  top,  Fig.  65,  about  OV  tends  to  increase  the  angle  (j>  independently  of  the 
torque  due  to  the  force  w^,  and  therefore  the  precessional  motion  of  OP  is  more 
rapid  than  would  be  produced  by    mg-  alone  if  the  top  were  symmetrical  about    O  V. 


ISO 


ELEMENTS   OF   MECHANICS. 


OP,  Fig.  66,  represent  the  angular  velocity  of  spin,  «,  of  a  wheel, 
and  let  O  be  the  angular  velocity  at  which  the  axis  OP  sweeps 
round  0.  The  line  OP  represents  co,  and  the  linear  velocity  of  the 
end  P  of  the  line  represents  the  angular  acceleration  a  which  is  in- 
volved in  the  co7ttiniial  change  of  direction  of  the  axis  OP.  There- 
fore, according  to  the  principles  enunciated  in  Art.  21,  we  have, 


a  =  0)11 


(i) 


when  the  axis  of  spin  is  at  right  angles  to  the  axis  of  precession, 
as  in  Fig.  64 ;  and 

a  =  0)12  sin  <^  (ii) 

when  the  axis  of  spin  makes  an  angle  ^  with  the  axis  of  preces- 
sion as  in  Fig.  65.  Using  these  values  of  a  in  equation  (32)  we 
have 

T=tDQ.K  (iii) 

when  the  axis  of  spin  is  at  right  angles  to  the  axis  of  precession ; 
and 

7"=  o)ftirsin  (^  (iv) 

when  the  axis  of  spin  makes  an  angle  ^  with  the  axis  of  preces- 
sion. 


Fig.  67. 


Fig.  68. 


The  above  analysis  of  the  action  of  the  gyroscope  will  hardly  be  convincing  to  the 
beginner  on  account  of  the  fact  that  the  action  is  analyzed  in  terms  of  the  rather 


ROTATORY   MOTION. 


151 


complicated  and  unfamiliar  ideas,  angular  velocity,  angular  acceleration,  moment  of 
inertia  and  torque;  it  is,  therefore,  desirable  to  analyze  the  action  of  the  gyroscope 
in  terms  of  the  fundamental  ideas  of  linear  velocity  and  acceleration,  mass,  and  linear 
force.  The  analysis  of  the  action  of  the  gyroscope  in  terms  of  linear  velocity  and 
acceleration  depends  upon  a  relation  which  is  sometimes  called  Coriolis'  law.  Given  a 
straight  tube  AB^  Fig.  67,  which  is  rotating  about  the  axis  C  at  angular  velocity  Q 
as  indicated  in  the  figure.  In  this  tube  is  a  ball  m  which  is  moving  away  from  C  at 
velocity  z/  (if  the  ball  were  moving  towards  C  its  velocity  v  would  be  considered  as 
negative).  Under  these  assumed  conditions  the  sidewise  acceleration^  a,  of  the  ball 
m  is  equal  to  2i2z/,  that  is 

a = 2Qv  (v) 

To  derive  this  relation,  the  sidewise  acceleration  a  may  be  considered  in  two  parts. 
In  the  first  place  we  have  the  acceleration  which  is  associated  with  the  continual 
change  of  direction  of  the  radial  velocity  v  of  the  ball.  This  accleration  is  equal  to 
Qv  as  shown  in  Fig.  68,  and  as  explained  in  Arts.  21  and  38.  In  the  second  place, 
as  the  ball  gets  farther  and  farther  away  from  the  axis  C,  its  actual  sidewise  velocity 
due  to  the  rotation  of  the  tube  increases,  but  this  sidewise  velocity  is  equal  to   flr, 


Fig.  69. 


according  to  equation  (29),  and  therefore,  since  v  is  the  rate  at  which  r  is  changing, 
it  is  evident  that  9.v  is  the  rate  at  which  the  sidewise  velocity,  Qr  is  changing,  as 
explained  in  Art.  20.  The  relation  a  =  2Q.V  is  used  in  the  discussion  of  the  motion 
of  steam  engine  governors,  where  the  governor  balls  have  a  motion  of  rotation  com- 
bined with  a  motion  towards  or  away  from  the  axis. 


152 


ELEMENTS   OF   MECHANICS. 


Consider  now  a  circular  disk  AB,  Fig.  69,  spinning  at  angular  velocity  o)  about 
its  axis  of  figure  O,  and  let  the  axis  O  be  turning  about  CD  at  angular  velocity  Q,. 
Consider  a  sample  particle  m  of  the  disk  at  a  distance  r  from  O  as  shown  in  Fig.  69. 
The  velocity  of  m  is  rw,  and  the  component  of  this  velocity  which  is  away  from 
the  axis  CD  is  rio  sin  6  ;  and,  therefore,  the  precessional  rotation  about  the  line 
CD  involves  an  acceleration  of  m  towards  the  reader  which,  according  to  equation 
(v),  is 

a  =  2wi2rsin^  (vi) 

It  may  be  easily  seen  that  this  acceleration  is  towards  the  reader  in  quadrants  i 
and  2,  and  away  from  the  reader  in  quadrants  3  and  4,  and,  therefore,  the  forces  re. 
quired  to  produce  these  accelerations  constitute  a  torque  about  the  axis  EF  as  indi. 
Gated  by  the  arrow  T. 

73.  Examples  of  precessional  rotation,  (a)  The  precession  of 
the  earth's^  axis.  —  The  attraction  of  the  sun  keeps  the  earth  in 
its  orbit.     The  force  of  attraction  of  the  sun  upon  the  bulging 


I    sun    y. 


plane  ot  earth's  orbit 


Fig.  70. 


equatorial  portion  a,  Fig.  70,  is  greater  than  the  "  centrifugal 
force  "  due  to  orbital  motion  of  the  earth,  and  the  force  of  attrac- 
tion on  the  bulging  equatorial  portion  b  is  less  than  the  "  centri- 
fugal force."  Therefore,  the  earth  is  acted  upon  by  an  unbalanced 
torque  about  0  which  causes  the  earth's  axis  to  describe  a  cone 
about  the  line  (9  F  which  is  at  right  angles  to  the  plane  of  the 
earth's  orbit.  The  action  of  the  moon  is  here  ignored  for  the 
sake  of  simplicity. 

(J?)  A  coin  rolling  along  the  floor  is,  of  course,  rotating,  and  the 
instant  the  coin  begins  to  be  inclined  to  either  side,  the  unbal- 
anced torque  due  to  gravity  causes  a  precessional  movement 
of  the  axis  of  the  coin,  and  the  coin  describes  a  curved  path  in 
consequence  of  this  precession. 


ROTATORY   MOTION. 


153 


{c)  Rotati7ig  parts  of  machines  on  ship-board.  —  The  pitching 
and  rolling  of  a  vessel  at  sea  causes,  at  each  instant,  a  certain 
angular  velocity  11  of  the  axis  of  a  rotating  machine  part,  and 
an  unbalanced  torque  is  immediately  brought  into  existence. 
For  example,  when  a  steamer  turns  round,  the  propeller  and 
propeller  shaft  change  direction  continuously,  when  a  steamer 
rolls  the  axis  of  a  dynamo  armature  which  is  athwart  ship 
changes  its  direction  periodically,  when  a  steamer  pitches  the 
axis  of  the  propeller  and  propeller  shaft  changes  its  direction 
periodically.  In  the  case  of  a  steamer  driven  by  steam  turbines 
the  propeller  shaft  turns  at  high  speed  and  the  rotating  member 
of  the  turbine  is  quite  heavy,  so  that  the  pitching  motion  of  such 


dash-pot 


V/////////////^ 


W^y 


?jr«sz3=: 


W////////M 


Fig.  71. 


I 


a  vessel  would  produce  excessively  large  forces  at  the  bearings 
which  support  the  shaft.  A  turbine  torpedo-boat  of  the  British 
Navy  went  down  in  a  heavy  sea  in  1899  or  1 900,  being  probably 
broken  in  two  by  the  very  great  forces  produced  by  the  pitching 
of  the  boat,  and  the  consequent  angular  motion  of  the  propeller 
shaft,  forces  which,  perhaps,  were  not  duly  considered  in  the  de- 
signing of  the  hull  and  supporting  structure  of  the  shaft. 

{d)    When  a  locomotive  turns  a  curve  the  wheels  and  axles  turn 


154 


ELEMENTS   OF   MECHANICS. 


about  a  vertical  axis,  and  a  torque  is  brought  into  existence. 
When  a  side- wheel  steamboat  turns,  the  precession  of  the  heavy 
side-wheels  and  shaft  causes  the  boat  to  list. 

{e)  The  use  of  the  gyroscope  (or  gyrostat  as  it  is  sometimes  called), 
for  preventing  the  rolling  of  a  ship  at  sea.     A  rapidly  rotating 

wheel  is  hung  from  a  hinge  so  that 
its  axis  may  swing  back  and  forth 
in  a  vertical  plane,  including  the 
keel  of  the  boat,  as  shown  in  Fig. 
7 1 .  The  lower  end  of  the  axis  is 
attached  to  a  rod  which  connects 
to  a  piston  in  what  is  called  a  dash- 
pot.  When  the  vessel  rolls  about 
the  keel  as  an  axis,  the  axis  of  the 
gyrostat  oscillates  back  and  forth, 
and  the  effect  of  the  friction  of 
the  piston  in  the  dash-pot  is  the 
same  as  if  the  rolling  of  the  ship 
were  hindered  by  excessive  friction, 
and  thereby  the  motion  of  rolling 
is  greatly  reduced.  A  small  Ger- 
man torpedo  boat,  115  feet  long  by 
1 2  feet  beam,  was  recently  equipped 
with  a  gyrostat  arranged  as  shown 
in  Fig.  71.*  The  gyrostat  wheel 
was  3.3  feet  in  diameter,  it  had  a 
mass  of  1,100  pounds  and  it  was 
driven  at  a  speed  of  1,600  revolu- 
tions per  minute.  The  effect  of 
this  arrangement  is  shown  in  Fig. 
72,  in  which  abscissas  measured  from  the  Hne  marked  0°  repre- 
sent angular  amplitudes  of  rolling  oscillations.  Above  the  point 
A  the  curve  shows  the  rolling  when  the  gyrostat  is  inoperative, 

*  See  paper  by  Otto  Schlick,  translated  in  Scientific  American  Supplement^  for 
January  26,  1907. 


20' 

/O'  2fi^ 

III 

tt" 

^ 

t: 

<:$ 

r\ 

0 

fi 

nffi;:   §:: 

if  1 

1 

Fig.  72. 


ROTATORY   MOTION. 


155 


and  below  the  point  A  the  curve  shows  the  rolling  when  the  gyro- 
stat is  in  action, 

KINEMATICS    OF   A    RIGID   BODY.* 

74.  Motion  of  a  rigid  body  in  a  plane.  —  A  rigid  body  is  said  to  move  in  a 
plane  when  all  points  of  the  body  which  lie  in  the  plane  remain  in  it.  For  example, 
a  rotating  wheel  moves  in  a  plane,  the  connecting  rod  of  a  steam  engine  moves  in  a 


Fig.  73. 


Fig.  74. 


plane.  Consider  a  rigid  body  AB,  Fig.  73,  moving  in  the  plane  of  the  paper.  The 
position  of  the  body  is  completely  indicated  by  the  position  of  the  line  AB  fixed  in 
the  body.     This  line  is  called  the  index  line. 

After  any  change  in  the  position  of  a  rigid  body  moving  in  a  plane,  a  certain  line 
in  the  body,  perpendicular  to  the  plane,  is  in  its  initial  position,  and  the  given  displace- 
ment is  equivalent  to  a  rotation  about  that  line  as  an  axis.  Let  AB  and  A^B^, 
Fig.  74,  be  the  positions  of  the  index  line  before  and  after  the  displacement.  Join 
AA^  and  BB^.  Erect  perpendiculars  from  the  middle  points  of  AA^  and  BB^ 
intersecting  at  p.  From  the  similarity  of  the  triangles  pAB  and  pA^B^  it  is  evi- 
dent that  the  same  part  of  the  body  is  at  /  before  and  after  the  displacement,  and 
that  the  line  through  /  perpendicular  to  the  paper  is  the  line  about  which  the  body 
may,  by  simple  rotation,  move  from  its  initial  to  its  final  position.  The  angle  A^ 
of  this  rotation  is  the  angle  subtended  by   AA^   or   BB^   as  seen  from  /. 

75.  The  instantaneous  motion  of  a  rigid  body  moving  in  a  plane  in  any  man- 
ner, is  a  motion  of  rotation  about  a  definite  line  called  the  instantaneous  axis  of  the 
motion.  Let  the  displacement,  shown  in  Fig.  74,  be  that  which  takes  place  in  a  short 
interval  of  time  A/;  then  A(p/At  is  the  instantaneous  angular  velocity  of  the  body, 
and  the  line  through  /,  perpendicular  to  the  paper,  is  the  instantaneous  axis.     During 

*  The  discussion  of  the  dynamics  of  a  rigid  body  should  properly  be  preceded  by  a 
discussion  of  the  kinematics  of  a  rigid  body.  This,  however,  has  not  been  done  be- 
cause most  of  the  discussion  of  the  dynamics  of  a  rigid  body  can  be  based  upon  the 
simple  idea  of  rotation  about  a  fixed  axis.  Thus  the  rotatory  motion  of  a  rolling 
wheel  is  in  no  way  different  from  what  it  would  be  if  the  translatory  motion  did  not 


156 


ELEMENTS   OF   MECHANICS. 


a  finite  interval  of  time  the  motion  of  a  body  may  be  irregular,  but  the  motion  of  a 
body  during  an  interval  of  time  approaches  uniformity  as  that  interval  approaches 
zero.  Therefore  the  motion  of  a  body  during  a  short  interval  of  time  is  the  simplest 
motion  which  can  produce  the  actual  displacement  which  occurs  during  the  interval. 

76.  Composition  of  angular  and  linear  displacements.  Consider  an  angular 
displacement  Af  of  a  body  about  the  point/,  Fig.  75,  bringing  the  point  O  to  O^  ; 
and  the  linear  displacement  A/  parallel  and  equal  to  0^0,  bringing  0^  back  to  O. 
These  two  displacements  are  together  equivalent  to  an  angular  displacement  A^  about 
O,  bringing  Op  to  (9/'.  Let  the  distance  of  /  from  the  line  00^  he  r\  then,  if  A0 
is  small,  Ll=:.  rA(p. 

77.  Resolution  of  motion  in  a  plane. —  From  Arts.  75  and  76  it  follows  that  the 
instantaneous  motion  of  a  rigid  body  in  a  plane  may  be  resolved  into  a  motion  of  rota- 
tion about  an  arbitrary  point  combined  with  a  certain  linear  velocity.  Consider  the 
actual  displacement  represented  in  Fig.  75 ,  namely,  a  rotation  about  O  bringing  /  to 
/'.     This  displacement  is  equivalent  to  an  equal  angular  displacement  A^,  about  the 


Fig.  75. 


Fig.  76. 


arbitrary  point/,  together  with  the  linear  displacement  0^0  ox  pp^ .  Let  this  linear 
displacement  be  Ax,  and  let  A/  be  the  interval  which  elapses  during  the  displace- 
ment. The  actual  angular  velocity  A0/A/  about  the  point  O  (the  instantaneous  cen- 
ter) is  equivalent  to  an  angular  velocity  ^(pj^-t  about  the  point  /  combined  with  a 
linear  velocity    Lxj^i   parallel  to//''. 

78.  Motion  of  a  rigid  body  with  one  point  fixed.  —  If  a  rigid  body,  one  point 
of  which  is  fixed,  is  displaced  in  any  manner  whatever,  a  certain  line  in  the  body  will 
be  in  its  initial  position  after  the  displacement,  and  the  given  displacement  will  be 
equivalent  to  a  rotary  movement  about  this  line  as  an  axis. 

Proof.  —  Consider  a  spherical  shell  of  the  body  having  its  center  at  a  fixed  point. 
Let  ABy  Fig.  76,  be  an  arc  of  a  great  circle  on  this  spherical  shell ;  the  position  of 
AB  fixes  the  position  of  the  body,  and  AB  is  called  the  index  line.  Let  the  move- 
ment of  the  body  bring  AB  io  A'B^.  Connect  AA^  and  BB'  by  arcs  of  great 
circles.  Draw  great  circles  bisecting  A  A'  and  BB'  at  right  angles.  The  point  / 
at  the  intersection  of  these  circles  bisecting  A  A'  and  BB'  has  the  same  position  rela- 
tive to   AB   and   A^ B' ,  so  that  this  point  of  the  shell  is   in  its  initial  position,  and 


ROTATORY   MOTION. 


157 


the  line  drawn  from  the  center  of  the  spherical  shell  to  the  point  /  is  the  axis  about 
which  the  given  movement  can  be  produced  by  rotation. 

The  instantaneous  motion  of  a  rigid  body  about  a  fixed  point  is  a  motion  of  simple 
rotation  at  definite  angular  velocity  about  a  definite  line  called  the  instantaneous  axis 
of  the  motion. 

79.  Vector  addition  of  angular  velocities.  — Consider  an  angular  velocity  about 
the  axis  a,  Fig.  77,  and  another  angular  velocity  about  the  axis  b ;  the  two  angular 


FlK.  77. 


Fig.  78. 


velocities  are  together  equivalent  to  an  angular  velocity  about  the  axis  r,  the  respective 
angular  velocities  being  proportional  to  the  lengths  of  the  lines  a,  b  and  c. 

Outline  of  proof .  —  Imagine  a  sphere  constructed  with  its  center  at  6>,  Fig.  77, 
and  let  a  and  b^  Fig.  78,  be  the  points  where  the  lines  a  and  b,  Fig.  77,  cut  the 
sphere.  Imagine  a  very  small  rotation  Aa  about  Oa  followed  by  a  very  small 
rotation   A^   about     Ob,    bringing  the  great  circle    ab   to  the  position    a^b^.     The 


Fig.  79. 


point  of  intersection  of  ab  and  a'F  is  the  point  where  the  resultant  axis  Oc  cuts  the 
sphere,  and  the  angle  Ar  is  the  amount  of  rotation  about  Oc  which  is  equivalent  to 
the  two  rotations  Aa  and  A^  ;  then  it  can  be  shown  that  the  three  angles  Aa,  A3 
and  A^  are  related  to  each  other  as  the  lengths  of  the  three  lines  a,  b  and  c.  Fig.  77, 
and  that  the  two  arcs   ac   and    cb    are  equal  to  ^  and  0  respectively  of  Fig.  77. 

80.  Vector  addition  of  torques.  —  Let  the  lines  a  and  b.  Fig.  79,  represent  two 
given  torques  and  let  it  be  required  to  show  that  a  and  b  are  together  equivalent  to 


15^  ELEMENTS   OF   MECHANICS. 

the  torque  c.  Draw  the  lines  1-2,  2-3,  and  1-3  perpendicular  to  and  bisected  by 
«,  b  and  c  respectively.  The  lengths  of  these  lines  are  proportional  to  the  lengths 
a^  b  and  c.  Imagine  the  torque  a  to  be  due  to  a  unit  upward  force  at  i  and  a  unit 
downward iorce.  at  2  (upward  and  downward  being  perpendicular  to  the  plane  of  the 
paper),  then  the  torque  b  is  equivalent  to  a  unitt  of  upward  force  at  2  and  a  unit  of 
downward  force  at  3;  but  the  upward  force  and  downward  force  at  2  annul  each  other, 
so  that  we  have  left  only  a  unit  of  upward  force  at  i  and  a  unit  of  downward  force 
at  3,  which  give  a  torque  about  the  line  c  proportional  to  the  length  of  c. 

Problems. 

86.  A  body  starts  from  rest  and  after  10  seconds  it  is  rotating 
5  5  revolutions  per  second.  What  is  the  average  angular  accel- 
eration ?     Express  the  result  in  radians  per  second  per  second. 

87.  In  what  terms  is  moment  of  inertia  expressed  :  (a)  When 
length  is  expressed  in  centimeters  and  mass  in  grams  ?  (U)  When 
length  is  expressed  in  inches  and  mass  in  pounds  ?  (c)  When 
length  is  expressed  in  feet  and  mass  in  pounds  ?  The  unit 
moment  of  inertia  in  case  {a)  is  the  c.  g.  s.  unit.  How  many 
c.  g.  s.  units  of  moment  of  inertia  are  there  in  the  unit  involving 
the  inch  and  the  pound,  and  in  the  unit  involving  the  foot  and 
the  pound  ? 

88.  Calculate  the  moment  of  inertia  of  a  uniform  slim  rod, 
length  3.1  feet  (=  /)  and  mass  3.6  pounds  (=  nt),  about  an  axis 
passing  through  the  center  of  the  rod  and  at  right  angles  to  the 
length  of  the  rod. 

(a)  Calculate  iTfrom  the  formula  K—  ml'^j  12. 

{b)  Calculate  K  approximately  by  multiplying  the  mass  of  each 
o.  I  foot  of  the  rod  by  the  square  of  its  estimated  mean  distance 
from  the  center  of  the  rod. 

(c)  Calculate  the  radius  of  gyration  of  the  rod. 

89.  Calculate  the  moment  of  inertia  of  a  circular  disk,  radius 
1.7  feet,  mass  4.25  pounks,  about  the  axis  of  figure. 

{a)  Calculate  Kfrom  the  formula  given  in  the  table  in  Art.  62. 
The  circular  disk  is  of  course  a  very  short  cylinder. 

(b)  Calculate  the  radius  of  gyration  of  the  disk. 

90.  {a)  Calculate  the  moment  of  inertia  of  the  rod,  problem  88,  about  an  axis 
passing  through  the  end  of  the  rod  and  perpendicular  to  the  rod. 


1^ 


ROTATORY   MOTION.  1 59 

(<^)  Calculate  the  moment  of  inertia  of  the  disk,  problem  89,  about  an  axis  pass- 
ing through  the  edge  of  the  disk  parallel  to  the  axis  of  figure  of  the  disk. 

91.  A  circular  disk,  5  feet  diameter,  weighing  1,200  pounds  is 
mounted  upon  a  shaft  6  inches  in  diameter.  The  disk,  set  rotating 
at  500  revolutions  per  minute  and  left  to  itself,  comes  to  rest  in 
75  seconds.  Calculate  average  (negative)  angular  acceleration 
while  stopping,  calculate  average  torque  acting  to  stop  the  disk, 
and  calculate  the  frictional  force  at  the  circumference  of  the  shaft. 

92.  What  is  the  kinetic  energy  of  the  disk  specified  in  problem 
91  when  the  speed  is  500  revolutions  per  minute? 

93.  A  metal  disk  12  inches  in  diameter  and  weighing  25 
pounds,  has  a  cylindrical  hub  projecting  on  each  side.  Each  hub 
is  I  inch  in  diameter  and  weighs  ^^  of  a  pound  (total  mass  25.5 
pounds).     What  is  the  moment  of  inertia  of  the  whole  ? 

The  hubs  of  this  disk  roll  on  a  track  which  drops  i  inch  ver- 
tically in  each  foot  of  horizontal  distance,  find  how  fast  the  disk 
gains  linear  velocity  in  rolling  down  this  track. 

94.  A  slim  rod  2  feet  long  and  weighing  2.5  pounds  is  sus- 
pended by  a  wire.  The  wire  is  attached  to  the  middle  of  the  rod 
and  the  rod  hangs  in  a  horizontal  position.  The  rod,  set  vibra- 
ting about  the  wire  as  an  axis,  makes  50  complete  vibrations  in 
10  minutes,  25  seconds.  What  torque  would  be  required  to 
twist  the  wire  through  one  complete  turn  ? 

95.  An  irregular  body  is  suspended  by  the  same  wire  that  is 
specified  in  problem  94,  and,  set  vibrating  about  the  wire  as  an 
axis,  it  makes  37  complete  vibrations  in  10  minutes.  What  is  its 
moment  of  inertia  ? 

96.  A  uniform  slim  rod  4  feet  long  is  hung  as  a  gravity  pen- 
dulum at  a  point  distant  6  inches  from  the  end  of  the  bar.  Cal- 
culate its  equivalent  length  as  a  pendulum. 

97.  A  pendulum  clock  rated  in  Boston  and  carefully  trans- 
ported to  Hammerfest  would  gain  how  many  seconds  per  day  ? 

See  table  in  Art.  yo. 

98.  The  connecting  rod  of  a  steam  engine  weighs  50  pounds, 
its  center  of  mass  is  distant  1 8  inches  from  the  center  of  the  hole 


l6o'  ELEMENTS   OF   MECHANICS. 

which  fits  the  crank  pin,  and  when  it  is  swung  as  a  gravity  pen- 
dulum about  the  point  a^  Fig.  98/,  it  makes  90  complete  vibra- 


^— ■t-:"-::g^ 


^ 


18  inches  — 

I  ■  I 

Fig.  98>. 

tions  in  one  minute.  The  diameter  of  the  hole  is  i^  inches. 
What  is  the  moment  of  inertia  of  the  connecting  rod  about  its 
center  of  mass  ? 

99.  A  stick  four  feet  long  and  weighing  10  pounds  is  held  ver- 
tically and  struck  a  horizontal  blow  with  a  hammer  at  a  point  18 
inches  from  the  upper  end,  which  is  released  at  the  instant  of  the 
blow.  The  impulse  of  the  blow  is  80  pounds-weight-seconds. 
Find  the  translatory  velocity  imparted  to  the  stick,  find  the  an- 
gular velocity  about  its  center  of  mass,  and  find  the  position  of 
the  point  in  the  stick  which  remains  stationary  for  a  moment 
after  the  hammer  blow. 

100.  Find  the  distance  from  the  axis  of  suspension,  of  the  slim 
rod  described  in  problem  96,  to  the  point  where  the  rod  may  be 
struck  horizontally  with  a  hammer  without  causing  a  side  force 
to  be  exerted  on  the  ax:s  of  suspension.  Compare  this  distance 
with  the  "  equivalent  length  "  of  the  rod  as  a  gravity  pendulum. 

101.  A  water  wheel  is  connected  to  its  belt  pulley  by  a  shaft. 
Find  the  torque,  in  pound-feet  and  in  pound-inches,  tending  to 
twist  the  shaft  when  the  water  wheel  develops  200  horse-power 
at  a  speed  of  600  revolutions  per  minute. 

A^oU.  —  The  two  equations  IV=  T(j)  and  /*  =  7b  correspond  exactly  to  equa- 
tions (24)  and  (25)  as  explained  in  Art.  66.  The  only  difficulty  involved  in  the  use 
of  the  equations    fV^  T(j)    and    P=  Tu   is  to  keep  the  units  straight,  as  it  were. 

102.  An  electric  motor,  running  at  900  revolutions  per  minute, 
develops  1 5  horse-power.     Find  the  torque  with  which  the  field 


ROTATORY   MOTION.  1 6 1 

magnet  acts,  upon  the  rotating  armature,  neglecting  friction.     Ex- 
press the  result  in  pound-feet  and  in  pound-inches. 

103.  The  armature  shaft  of  a  ship's  dynamo  is  athwart  ship, 
and  the  armature  is  driven  clockwise  as  seen  from  the  port  side 
of  the  vessel.  Describe  accurately  the  forces  with  which  the 
bearings  act  upon  the  armature  shaft  as  the  vessel  rolls.  Specify 
the  directions  of  these  forces  when  the  port  side  of  the  vessel  is 
rising,  and  when  the  port  side  of  the  vessel  is  falling. 

This  problem  refers  to  the  forces  which  arise  from  the  rotatory 
motion  of  the  armature.  The  port  side  of  a  vessel  is  on  the  left 
hand  of  a  person  facing  the  bow. 

104.  A  side-wheel  steamboat  is  suddenly  turned  to  port,  and 
the  gyrostatic  action  of  the  paddle  wheels  causes  the  boat  to  list. 
In  which  direction  does  the  boat  list,  to  starboard  or  port  ?    Why  ? 

105.  The  vessel  described  in  problem  104  is  steered  in  a  circle 
150  feet  in  radius  at  a  velocity  of  25  feet  per  second,  and  the 
vessel  lists  5°  because  of  the  gyrostatic  action  of  the  paddle 
wheels  and  shaft.  To  produce  a  5°  Hst  when  the  boat  is  stand- 
ing still  requires  a  weight  of  10  tons  to  be  shifted  from  the  center 
of  the  boat  to  a  point  1 5  feet  from  the  center.  The  paddle 
wheels  make  75  revolutions  per  minute.  Find  the  moment  of 
inertia  of  the  axle  and  wheels. 

106.  A  locomotive  rounds  a  curve  of  radius  528  feet  at  a  speed 
of  30  miles  per  hour.  The  diameter  of  the  driving  wheels  is  6 
feet  and  each  pair  of  drivers  and  the  connecting  axle  has  a  moment 
of  inertia  of  37000  pound-feet  ^.  Find  the  torque  acting  on 
each  pair  of  drivers  due  to  the  precession.  How  does  this  pre- 
cession modify  the  force  with  which  the  wheels  push  on  the  two 
rails  ? 

107.  A  torpedo  boat  makes  a  complete  turn  in  84  seconds 
and  its  propeller  rotates  at  a  speed  of  270  revolutions  per 
minute.  The  moment  of  inertia  of  the  propeller  is  2000  pound- 
feet  ^.  Required  the  precessional  torque  on  the  propeller  shaft. 
In  what  direction  does  this  torque  tend  to  bend  the  shaft  ? 

108.  A  high  speed  engine  with  its  shaft  athwart  ship,^  makes  240 


l62  ELEMENTS   OF   MECHANICS. 

revolutions  per  minute.  The  rim  of  the  fly-wheel  has  a  radius 
of  3  feet  and  a  mass  of  600  pounds.  Calculate  the  moment  of 
inertia  of  the  wheel  (rim).  The  maximum  angular  velocity  at- 
tained by  the  vessel  in  rolling  is  J^  radian  per  second.  Calculate 
the  maximum  torque  acting  on  the  fly-wheel  shaft  due  to  gyro- 
static  action  and  specify  the  direction  of  the  torque. 


CHAPTER   VIII. 
ELASTICITY  (STATICS). 

81.  Stress  and  strain.  — When  external  forces  act  upon  a  body 
and  tend  to  change  its  shape,  the  body  is  distorted  more  or  less, 
and  the  external  forces  are  balanced  by  the  tendency  of  the  dis- 
torted body  to  return  to  its  original  shape.  The  distortion  of  a 
body  always  brings  forces  into  action  between  the  contiguous 
parts  of  the  body  throughout.  These  force  actions  between  con- 
tiguous parts  of  a  distorted  body  are  called  internal  stresses  ;  and 
the  total  reaction  of  the  distorted  body,  which  balances  the  ex- 
ternal distorting  force,  is  called  the  integral  stress  of  the  body. 

The  actual  movement  of  the  point  of  application  of  an  external 
force  which  distorts  a  body  is  called  the  integral  strain  of  the 
body,  and  the  change  of  shape  of  each  small  part  of  the  distorted 
body  is  called  the  internal  strain.  Thus,  the  elongation  of  a  wire 
under  tension,  the  shortening  of  a  column  under  compression, 
the  angular  movement  of  the  end  of  a  rod  under  torsion,  the  de- 
pression of  the  middle  of  a  beam  which  is  loaded  at  its  center, 
and  the  decrease  of  volume  of  a  body  which  is  subjected  to 
hydrostatic  pressure  are  integral  strains ;  and  the  total  stretch- 
ing force  acting  on  the  wire,  the  total  load  on  the  column,  the 
total  torque  tending  to  twist  the  rod,  the  total  load  at  the  middle 
of  the  beam,  and  the  hydrostatic  pressure  which  acts  on  a  body, 
are  integral  stresses.  In  each  of  these  cases,  furthermore,  each 
small  part  of  the  body  is  distorted,  and  force  actions  exist  be- 
tween contiguous  parts  of  the  body  throughout.  These  are 
called  the  internal  strains  and  the  internal  stresses  respectively. 

82.  Homogeneous  and  non-homogeneous  stresses  and  strains.  — 
It  is  generally  the  case  in  a  distorted  body,  that  each  small  part 
of  the  body  is  differently  distorted,  and  that  the  internal  stress 
varies  from  point  to  point  in  the  body.     For  example,  the  dif- 

163 


164 


ELEMENTS   OF   MECHANICS. 


ferent  parts  of  a  bent  beam,  or  of  a  twisted  rod,  are  differently  dis- 
torted, and  the  internal  stress  varies  from  point  to  point ;  the 
pressure  of  the  atmosphere  decreases  and  the  air  becomes  less 
and  less  dense  with  increasing  altitude  above  the  level  of  the 
sea  ;  the  pressure  at  a  point  in  a  body  of  water  increases  with  the 
depth  beneath  the  surface,  and  the  water  is  more  and  more  com- 
pressed as  the  pressure  increases  ;  the  stress  in  a  long  cable, 
which  is  suspended  in  a  mine  shaft,  increases  from  the -lower  end 
upwards,  and  the  extent  to  which  each  portion  of  the  cable  is 
stretched  increases  with  the  stress. 

When  the  force  action  between  contiguous  parts  of  a  body  is 
the  same  at  every  point  in  the  body,  the  stress  is  said  to  be 


wire  under 
tension 


column  under 
compression 


body  A  under 
hydrostatic  pressure 


Fig.  80. 


homogeneous ;  and  when  every  part  of  a  body  is  similarly  dis- 
torted, the  strain  is  said  to  be  homogeneous.  Thus  each  part  of 
a  rod  under  tension  or  compression  is  similarly  distorted  as  shown 
in  Figs.  '^6a  and  Z6b^  and  the  force  action  or  stress  is  the  same  at 
every  point  as  shown  in  Figs.  %^a  and  85^^;  the  steam  in  a  boiler 
is  under  the  same  pressure  throughout  (gravity  negligible),  and 
the  degree  of  compression  of  every  portion  of  the  steam  is  the 
same ;   the  water  in  the  high  pressure  cylinder  of  a  hydraulic 


ELASTICITY. 


165 


press  is  under  the  same  pressure  throughout  (gravity  negligible), 
and  the  degree  of  compression  of  every  portion  of  the  water  is 
the  same. 


Fig.  81. 


The  distinction  between  homogeneous  and  non-homogeneous 
strains  is  shown  in  Figs.  81,  82,  and  83,  which  are  photographs 


r~> 

^   c_^ 

c~:> 

c 

'3 

--.-- 

C3^\C> 

C3 

ci:> 

c 

I>, 

0      \ 

( <o> 

CD 

CD 

-  C3 

CD  j 
CD        J 

C_>- 

C2> 

D        / 

"D 

M 

■igHi 

■ 

g 

iJHHB 

Fig.  82. 

of  a  thin  rubber  sheet  upon  which  a  large  circle  and  a  number  of 
small  circles  were  drawn.     Figure  81  shows  the  unstrained  sheet, 


1 66 


ELEMENTS   OF   MECHANICS. 


Fig.  82  shows  the  sheet  homogeneously  strained,  and  Fig.  83 
shows  the  sheet  non-homogeneously  strained.  The  small  circles 
are  changed  to  ellipses  in  Fig.  82  and  in  Fig.  83,  but  in  Fig.  82 
the  ellipses  are  all  alike  and  their  axes  are  in  the  same  direction, 
whereas,  in  Fig.  83,  some  of  the  ellipses  are  more  elongated  than 
others  and  their  axes  are  not  parallel.  In  a  homogeneous  strain 
a  /arg-e  portion  of  a  substance  is  distorted  in  a  manner  exactly 
similar  to  the  distortion  of  each  S7na//  portion  of  the  substance. 
This  is  shown  in  Fig.  82  in  which  the  large  ellipse  is  exactly  the 
same  shape  as  the  small  ones.  In  a  non-homogeneous  strain  a 
/a7^£-e  portion  of  a  substance  is  irregularly  distorted.      This  is 


■    /o      0       '^0   1 

0          0       „0                        q/ 

/           ---■         0  / 

I      0                  ■         Q      / 

1 

/ 

C      , --. 

f^       > 

Fig.  83. 


shown  in  Fig.  83  by  the  irregular  curve  into  which  the  large 
circle  has  been  converted  by  the  strain,  /t  is  a  fact  of  fimda- 
mental  importance  in  the  theory  of  elasticity  ^  that,  however  irregularly 
a  body  may  be  distorted,  any  small  portion  of  the  body  suffers  that 
simple  kind  of  distortion  which  changes  a  sphere  into  an  ellipsoid, 
or  which,  in  the  case  of  a  thin  sheet  of  rubber,  changes  a  circle  into 
an  ellipse.     That  is,  the  change  of  shape  of  any  small  portion  of 


ELASTICITY. 


67 


a  distorted  body  consists  of  an  increase  or  decrease  of  linear 
dimensions  in  three  mutually  perpendicular  directions,  and,  in 
some  cases,  this  simple  kind  of  distortion  is  accompanied  by  a 
slight  rotation  of  the  small  parts  of  the  body.  Thus,  in  Fig.  90, 
which  represents  a  portion  of  a  bent  beam,  the  short  straight 
lines  were  all  horizontal  or  vertical  in  the  unbent  beam. 

The  effect  of  a  sharp  groove  in  a  body  which  is  under  stress  is 
a  matter  of  very  great  practical  importance.  The  effect  is,  in 
general,  to  produce  an  excessive  concentration  of  stress  in  the 
material  at  the  bottom  of  the  groove,  and  a  crack  or  fracture  is 
almost  sure  to  develop,  unless  the  material  is  plastic  so  that  the 
bottom  of  the  groove  is  broadened  by  yielding.  Consider,  for 
example,  a  beam  in  which  a  sharp  groove  is  cut,  as  shown  in 
Fig.  84.     The  fine  lines  in  this  figure  represent  the  lines  of  stress 


I 


Fig.  84. 


when  the  beam  is  bent,  and  the  crowding  together  of  these  lines 
as  they  pass  under  the  groove  represents  the  concentration  of 
stress  above  referred  to. 

The  most  striking  illustration  of  the  effect  of  a  sharp  groove  in 
a  body  under  stress  is  furnished  by  a  piece  of  glass  in  which 
there  is  a  minute  crack.  A  piece  of  glass  without  a  crack  will 
stand  a  very  considerable  stress,  but  if  the  stress  "  flows  "  round 
the  end  of  a  crack,  the  stress  is  concentrated  and  the  crack 
extends  indefinitely.  A  pane  of  window  glass  or  a  glass  tumbler 
is  worthless  when  a  crack  once  starts.  A  less  familiar  illustra- 
tion of  the  effect  of  a  sharp  groove  is  furnished  by  the  method 
commonly  employed  for  breaking  a  bar  of  steel ;  thus,  a  steel 
rail,  which  normally  withstands  the  tremendous  stresses  due  to 


1 68.  ELEMENTS   OF   MECHANICS. 

the  weight  of  a  locomotive,  can  be  broken  in  two  by  a  hammer 
blow  if  a  nick  is  made  across  the  top  of  the  rail  with  a  sharp 
chisel.  Sharp  re-entrant  angles  are  always  carefully  avoided  in 
the  designing  of  those  parts  of  structures  which  are  intended  to 
sustain  stress. 

83.  Solids  and  fluids.  —  Everyone  is  familiar,  in  a  general  way, 
with  the  three  classes  of  substances,  solids,  liquids,  and  gases. 
A  solid  can  withstand,  for  an  indefinite  length  of  time,  a  stress 
which  tends  to  change  its  shape.  A  solid  which  recovers  from 
distortion  (strain)  when  stress  ceases  to  act,  is  said  to  be  elastic. 
Thus  good  spring  steel  recovers  almost  completely  from  a  moder- 
ate amount  of  distortion  when  the  distorting  force  (stress)  ceases 
to  act.  A  solid  which  does  not  recover  from  strain  when  the 
stress  ceases  to  act,  is  said  to  be  plastic.  Thus  lead  and  wax  are 
plastic  solids.  No  solid,  perhaps,  recovers  completely  from  dis- 
tortion ;  and,  on  the  other  hand,  every  plastic  substance,  perhaps, 
is  slightly  elastic.  Thus  the  best  spring  steel  does  not  completely 
recover  from  even  a  slight  distortion,  and  when  the  distortion  is 
great  the  steel  takes  a  very  decided  permanent  set ;  and  even  wax 
is  slightly  elastic,  as  is  shown  by  the  distinct  metallic  ring  of  a 
large  cake  of  bees  wax  or  paraffine  when  it  is  struck  with  a 
hammer. 

A  fluid  is  a  substance  which,  at  rest,  cannot  sustain  a  stress 
which  tends  to  change  its  shape.  While  a  fluid  is  actually  chang- 
ing shape,  however,  it  does  sustain  a  stress  which  tends  to  change 
its  shape.  Thus  a  stream  of  syrup  falling  from  a  vessel  is  under 
tension  like  a  stretched  rope,  and  the  effect  of  this  tension  is  to 
continually  lengthen  each  portion  of  the  stream  of  syrup  as  it 
falls.  A  fluid  at  rest  always  pushes  normally  against  every  por- 
tion of  a  surface  which  is  exposed  to  the  action  of  the  fluid. 
Thus  the  steam  in  a  boiler  pushes  outwards  on  the  boiler  shell 
at  each  point,  the  water  in  a  vessel  pushes  normally  against  the 
walls  of  the  vessel  at  each  point,  and  the  atmosphere  pushes 
normally  against  every  portion  of  an  exposed  surface.  A  fluid 
in  motion,  however,  may  not  push  normally  against  an  exposed 


ELASTICITY.  1 69 

surface.  Thus,  a  water  pipe  is  subject  only  to  a  bursting  force, 
if  the  water  is  at  rest ;  but  if  the  water  flows  through  the  pipe, 
it  has  a  slight  tendency  to  drag  the  pipe  along  with  it.* 

A  liquid  is  a  fluid,  like  water  or  oil,  which  can  have  a  free  sur- 
face, such  as  the  surface  of  water  in  a  glass.  A  gas^  on  the 
other  hand,  is  a  fluid  which  completely  fills  any  containing  vessel. 

84.  Hooke's  law.  Elastic  limit.  —  Robert  Hooke  discovered, 
in  1676,  that  what  we  have  called  the  integral  strain  of  a  body  is 
quite  accurately  proportional  to  what  we  have  called  the  integral 
stress,  t  Thus,  the  elongation  of  a  wire  under  tension  is  propor- 
tional to  the  stretching  force,  the  shortening  of  a  loaded  column 
is  proportional  to  the  load,  the  angular  movement  of  the  end  of 
a  rod  under  torsion  is  proportional  to  the  torque  which  acts  on 
the  rod,  and  the  depression  of  a  loaded  beam  is  proportional  to 
the  load. 

Elastic  limit.  —  Hooke's  law  is  quite  accurately  true  for  dis- 
tinctly elastic  substances  like  steel,  but  it  does  not  apply  to  plas- 
tic substances,  and  even  for  elastic  substances  like  steel  there  is 
a  limit,  called  the  elastic  limits  beyond  which  stress  and  strain  are 
no  longer  even  approximately  proportional.  When  an  elastic 
substance  is  strained  beyond  its  elastic  limit  it  does  not  return  to 
its  original  size  or  shape  when  the  stress  ceases  to  act,  but  takes 
what  is  called  a  permanent  set.  Liquids  and  gases,  however,  re- 
turn to  their  exact  initial  volume  when  relieved  from  pressure, 
provided  the  temperature  has  not  changed,  that  is,  liquids  and 
gases  may  be  said  to  be  perfectly  elastic,  but  when  a  liquid  or 
gas  is  compressed  the  diminution  of  volume  (integral  strain)  is 
not  proportional  to  the  increase  of  pressure  (integral  stress)  ex- 
cept when  the  increase  of  pressure  is  fairly  small.  This  is  at 
once  evident  in  the  case  of  gases  when  we  consider  that  they 
conform  to  Boyle's  law  as  explained  in  Art.  loi. 

*  A  jet  of  water  issuing  from  the  end  of  a  pipe  pushes  backwards  on  the  pipe,  as 
every  fireman  knows.  This  backward  force  is  due  to  the  normal  force  with  which  the 
water  pushes  on  the  inner  walls  of  the  pipe  where  the  pipe  bends. 

I  It  follows  from  this  experimental  fact  that  the  strain  at  each  point  of  a  distorted 
elastic  body  is  proportional  to  the  stress  at  that  point. 


170  ELEMENTS   OF   MECHANICS. 

86.  Limitations  and  plan  of  this  chapter.  —  The  phenomena 
which  are  associated  with  the  distortion  of  bodies  are  excessively 
compHcated.  Let  one  consider  the  swaying  of  objects  in  the 
wind,  the  bending  and  compression  of  structures  under  load  and 
their  vibration  with  sudden  variations  of  load ;  let  one  think  of 
all  the  familiar  properties  of  brittle  substances  like  chalk  and 
glass,  of  plastic  substances  like  clay  and  wax  and  of  elastic  sub- 
stances like  steel  and  rubber ;  let  one  consider  that  all  of  the 
phenomena  of  sound  are  due  to  the  vibrations  of  bodies,  and  to 
wave  movements  in  the  air,  and,  in  many  cases,  to  wave  move- 
ments in  water  and  in  solids,  all  of  which  have  to  do  with  distor- 
tion and  compression ;  and  let  one  think  that  local  changes  of 
shape  and  compression  and  expansion  are  involved  inextricably 
in  nearly  every  case  of  flow  of  air  and  water.  Let  one  think  of 
all  of  these  things  and  then  consider  whether  it  is  not  necessary 
to  bring  the  mind  to  some  narrow  view  before  any  clear  Une  of 
argument  can  be  pursued  relative  thereto ! 

Of  all  the  great  variety  of  solid  substances,  having  almost 
every  imaginable  degree  of  elasticity,  plasticity,  hardness,  and 
brittleness,  and  ranging  in  strength  from  sun  dried  clay  to  the 
toughest  steel,  we  are  here  concerned  only  with  the  behavior 
under  stress  of  those  which  are  used  as  materials  of  construction; 
and,  in  addition,  we  are  here  concerned  with  the  tendency  of 
increase  of  pressure  to  reduce  the  volumes  of  liquids  and  gases. 

A  substance,  like  wood,  which  has  a  grained  structure,  is  said 
to  be  aeolotropic  (pronounced  e'-o-lo-trop'-ic).  Most  crystalline 
substances,  and  rolled  and  drawn  metal  are  aeolotropic.  A  sub- 
stance, like  glass  or  water,  which  does  not  have  a  grained  struc- 
ture, is  said  to  be  isotropic.  The  behavior  under  stress  of  aeolo- 
tropic substances  is  very  complicated ;  these  complications  need 
not  be  considered,  however,  for  practical  purposes,  because  sub- 
stances having  a  grained  or  fibrous  structure  are  generally 
subjected  to  stresses  parallel  to  the  grain,  as  in  beams,  and 
ropes,  and  wires.  The  difference  between  aeolotropic  substances 
and  isotropic  substances  is  ignored  in  this  chapter. 


ELASTICITY.  171 

Types  of  stress  and  strain.  —  In  the  discussion  of  the  behavior 
of  bodies  under  stress,  it  is  necessary  to  consider  three  simple 
types  of  stress  and  strain.  Thus  we  have  longitudinal  stress  and 
longitudinal  strain^  which  is  the  type  of  stress  and  strain  in  a  rod 
under  tension  or  in  a  column  under  compression ;  we  have  hydro- 
static pressure  and  isotropic'^  strain^  which  is  the  type  of  stress 
and  strain  in  a  body  subjected  to  hydrostatic  pressure ;  and  we 
have  shearing  stress  and  shearing  strain,  which  is  the  type  of 
stress  and  strain  which  exists  (non-homogeneously)  in  a  twisted 
rod.  A  discussion  of  the  first  two  types  of  stress  and  strain  is 
sufficient  for  most  practical  purposes,  and,  therefore,  the  discus- 
sion of  shearing  stress  and  shearing  strain  is  given  in  small  type 
preceeding  the  outline  of  the  general  theory  of  stress  and  strain. 

Treatment  of  no7i-homogeneoiis  stresses  and  strains.  —  In  the 
following  discussion,  the  behavior  of  a  substance  under  each  type 
of  homogeneous  stress  and  strain,  is  first  considered,  and  the  ideas 
so  developed  are  used  as  a  basis  for  the  discussion  of  important 
cases  of  non-homogeneous  stresses  and  strains.  For  example, 
the  discussion  of  the  bent  beam  is  based  upon  the  discussion  of 
homogeneous  longitudinal  stress  and  strain,  and  the  discussion 
of  the  twisted  rod  is  based  upon  the  discussion  of  homogeneous 
shearing  stress  and  strain. 

LONGITUDINAL   STRESS   AND    STRAIN. 

86.  Longitudinal  stress. — ^^  Figure  85^  represents  a  portion  of 
a  rod  under  tension.  Let  F  be  the  total  force  tending  to  stretch 
the  rod,  and  let  A  be  the  sectional  area  of  the  rod ;  then  the 
stretching  force  per  unit  of  sectional  area  is  FjA  (=  P),  and  the 
force  action  between  contiguous  portions  of  the  rod  is  as  follows  : 
Imagine  a  horizontal  unit  of  area  q  anywhere  in  the  material  of 
the  rod,  the  material  on  the  two  sides  of  q  exerts  a  pull  P  across 
q,  as  shown  in  Fig.  85^^;  imagine  a  vertical  unit  of  area  q'  drawn 

*When  an  isotropic  substance,  such  as  glass,  is  subjected  to  a  hydrostatic  pressure 
the  substance  is  reduced  in  volume  without  being  changed  in  shape.  Such  a  strain  is 
called  an  isotropic  strain,  for  want  of  a  better  name. 


1/2 


ELEMENTS   OF   MECHANICS. 


anywhere  in  the  material  of  the  rod,  the  material  on  the  two  sides 
of  q'  does  not  exert  any  force  at  all  across  q' .  The  force  acting 
across  any  horizontal  area  of  a  units  is,  of  course,  equal  to  Pa. 
The  force  per  unit  area,  P,  is  the  measure"^  of  the  longitudinal 
stress,  and  the  direction  of  P  is  called  the  axis  of  the  stress. 

A  rod  under  tension  may  be  considered  as  a  case  of  positive 
longitudinal  stress,  and  a  rod  or  column  under  compression  may 
be  considered  as  a  case  of  negative  longitudinal  stress. 


rod  nnder  tension 

Fig.  85«. 


rod  under  compression 

Fig.  853. 


87.  Longitudinal  strain.  —  A  rod  under  tension  is  longer  than 
when  it  is  not  under  tension,  and,  since  each  unit  portion  of  the 
rod  must  be  equally  stretched,  it  is  evident  that  the  increase  of 
length  of  the  rod  is  proportional  to  its  total  initial  length.  There- 
fore it  is  most  convenient  to  express  the  increase  of  length  as  a 
fraction  of  the  total  initial  length  ;  thus,  a  stretch  of  2  thou- 
sandths means  an  increase  of  length  equal  to  2  thousandths  of 
the  total  initial  length  of  a  rod.  Let  L  be  the  initial  length  of 
a  rod  and  let  /  be  its  increase  of  length  under  tension,  then 
//L  (=yS)  is  the  increase  of  length  expressed  as  a  fraction  of  the 
initial  length,  and,  of  course,  the  ratio  of  L  -\- 1  to  Z  is  equal  to 


*  That  is,  the  number  which  is  used  to  specify  the  value  of  the  stress.     See  Art.  1 1. 


ELASTICITY. 


^7Z 


I  +  yS.  The  fraction  ^  (=  //L)  is  used  as  the  measure  of  the 
longitudinal  strain. 

The  character  of  the  distortion  of  the  parts  of  a  rod  under  ten- 
sion is  shown  in  Fig.  Z6a^  any  spherical  portion  of  the  material 
of  the  rod  becomes  an  ellipsoid  of  revolution.  The  major  axis 
of  the  ellipsoid  is  I  +  /3  times  the  diameter  of  the  original 
sphere. 

Lateral  contraction  of  a  stretched  rod.  Polsson's  ratio, — When 
a  rod  is  stretched,  it  contracts  laterally  as  indicated  by  the  dotted 


rod  nnder  tensiop 

Fig.  86a. 


rod  tinder  compression 
Fig.  86^. 


» 


ellipses  in  Fig.  ^6a.  This  lateral  contraction  of  a  stretched  body 
is  very  strikingly  shown  by  a  stretched  rubber  band.  The  lateral 
contraction  <^  of  a  stretched  rod  is  most  conveniently  expressed 
as  a  fraction  of  the  original  diameter  D  of  the  rod,  and  the  frac- 
tion dj  D  we  may  represent  by  the  letter  yS'.  The  elongation 
of  a  rod  per  unit  length  (/3)  bears  for  a  given  substance  a  fixed 
ratio  to  the  lateral  contraction  per  unit  of  diameter  (/3'),  and  this 
ratio  is  called  Polsson's  ratio ;  /3  is  approximately  four  times  as 
large  as  /3'  for  steel,  brass,  and  copper. 

88.  Stretch  modulus   of  a  substance.  —  The  elongation  of  a 
rod  under  tension  is  proportional  to  the  stretching  force  (Hooke's 


174  ELEMENTS   OF   MECHANICS. 

law).  Therefore  the  stretching  force  divided  by  the  elongation 
is  a  constant  for  a  given  rod,  and  if  we  divide  stretching  force 
per  unit  area^  P,  by  elongation  per  unit  of  original  length 
yS(=//Z),  the  result  is  a  constant  which  depends  upon  the 
material  of  the  rod,  but  which  is  independent  of  the  size  and 
length  of  the  rod.  This  constant,  which  is  represented  by  the 
letter  E,  is  called  the  stretch  modulus  *  of  the  material  of  which 
the  rod  is  made.     That  is 

£=|  (48) 

The  stretch  modulus  may  be  defined  in  a  slightly  different  way 
as  the  factor  which,  multiplied  by  the  elongation  per  unit  length, 
gives  the  stretching  force  per  unit  sectional  area  of  a  rod  under 
tension  {Efi  =  P) ;  and  it  is  evident  from  this  definition  that  E  is 
expressed  in  units  of  force  per  unit  of  area,  inasmuch  as  y8  is  a 
ratio  of  two  lengths  {I IE) ;  in  fact  E  is  the  force  per  unit  sec- 
tional area  which  would  double  the  length  of  a  rod  if  the  elonga- 
tion would  continue  to  be  proportional  to  the  stretching  force. 
This,  of  course,  is  not  true  for  elongations  of  more  than  a  few 
parts  per  thousand. 

89.  Determination  of  stretch  modulus. — The  stretch  modulus 
may  be  determined  by  applying  a  known  stretching  force  /^  to  a 
rod  of  known  length  L  and  known  sectional  area  A,  and  observ- 
ing the  increase  of  length  /.  Then  P^^FjA,  and  ^^IjL, 
so  that   E  =  Pj^  =  FLjAL     An  easier  method  for  determining 

TABLE. 
Values  of  the  stretch  modulus  of  various  substances. 
(In  pounds- weight  per  square  inch.) 
Copper  (drawn)  17,700,000 

Steel  (rolled)  29,800,000 

Wrought  iron  29, 600, 000 

Cast  iron  16,000,000 

Glass  9,600,000 

Oak  wood  1,450,000 

Poplar  wood  750,000 

♦Often  called  Young's  modulus,  or  **  the  modulus  "  of  elasticity  by  engineers. 


ELASTICITY. 


175 


the  stretch  modulus  of  a  substance  is  by  observing  the  deflection 
of  a  loaded  beam  as  explained  in  Art.  91. 

90.  Potential  energy  of  longitudinal  strain. — The  potential 
energy  per  unit  volume  of  a  stretched  (or  compressed  rod)  is 
equal  to  one  half  the  product  of  the  stress  P  and  the  strain  yS,  or 
it  is  equal  to  one  half  of  the  product  of  the  stretch  modulus  E 
and  the  square  of  the  strain  ^.     That  is 

W=  JP/S  (49) 


or 


W=\E^ 


(50) 


in  which  W  is  the  potential  energy  per  unit  of  volume  of  a 
stretched  (or  compressed)  rod,  P  is  the  stretching  (or  com- 
pressing) force  per  unit  sectional  area  of  the  rod ;  /8  is  the 
increase  (or  decrease)  of  length  per  unit  of  original  length, 
and  E  is  the  stretch  modulus  of  the  material.  If  P  and  E  are 
expressed  in  pounds-weight  per  square  inch,  W  is  expressed  in 
inch-pounds  of  energy  per  cubic  inch  of  material. 


Axis  otF 


Proof. — The  increase  of  length  /  of  a  rod  is  proportional  to  the 
stretching  force  F,  so  that  by  plotting  corresponding  values  of  / 
and  F  we  have  a  straight  line    OA,    as  shown  in  Fig.  ^y. 

Imagine  the  stretching  force  to  increase  slowly  from  zero  to  /% 
the  stretch  at  the  same  time  increasing  from  zero  to  /.  Let  F' 
be  an  intermediate  value  of  F,  and  let  A/  be  a  very  small  increase 
of  /  due  to  a  slight  increase  of  F' ,  then  F'  •  A/  is  the  work  done 


1/6 


ELEMENTS   OF   MECHANICS. 


on  the  rod  during  the  very  shght  increment  of  stretch  A/.  But 
F'  -A/  is  the  area  of  the  narrow  parallelogram  shown  in  Fig.  87, 
and  therefore  the  total  work  done  on  the  rod  while  the  stretching 
force  increases  from  zero  to  F  is  the  total  area  OAB  Fig.  87, 
which  is  equal  to  \Fl,  so  that  the  work  done  per  unit  volume  of 
the  rod  is    \FL   divided  by  the  volume   AL    of  the  rod.     That  is 


W=\ 


'AL 


4m 


This  proof  may  be  stated  in  a  slightly  different  way  thus  :  The 
average  value  of  the  stretching  force  between  zero  stretch  and  the 
given  stretch  /,  is  \F,  which,  multiplied  by  the  elongation  /, 
gives  the  work  done  on  the  rod. 

Equation  (50)  is  derived  from  equation  (49)  by  substituting 
Efi   for  P  according  to  equation  (48). 

91.  Discussion  of  a  bent  beam.*  —  The  simplest  case  of  a  bent 
beam  is  that  which  is  shown  in  Fig.  88  in  which  a  long  beam  is 
laid  horizontally  across  two  supports  55,  and  bent  by  hanging 
weights  on  the  projecting  ends  as  shown.  The  bending  moment 
or  torque  of  each  weight  is  equal  to  Wx,  and  this  bending  torque 
acts  on  every  portion  of  the  beam  between  the  supports  55,  so 
that  this  portion  of  the  beam  becomes  an  arc  of  a  circle.  All  of 
the  filaments  in  the  upper  part  of  the  beam  are  elongated,  all  of 

*  When  a  beam  is  bent  the  stretched  filaments  on  one  side  of  the  beam  contract 
laterally  and  the  compressed  filaments  on  the  other  side  of  the  beam  expand  laterally 
and  the  section  of  a  beam  originally  square  becomes  distorted  somewhat  as  shown  in 
Fig.  89.     This  is  very  clearly  shown  by  bending  a  rectangular  bar  of  rubber,  a  lead 

Fig.  89. 


Section  of  beam  before  bending. 


Section  of  beam  after  bending  (exaggerated). 


pencil  eraser  for  example.     This  distortion  of  the  section  of  a  bent  beam  is  usually 
very  slight  and  it  is  neglected  in  the  above  discussion. 


ELASTICITY. 


177 


the  filaments  in  the  lower  part  of  the  beam  are  shortened,  and 
certain  filaments  pp^  which  lie  in  what  is  called  the  median  line 
or  surface  of  the  beam,  remain  unchanged  in  length.     The  beam 


is  everywhere  under  longitudinal  strain  as  shown  in  Fig.  90 ;  and 
the  force  action  between  contiguous  parts  of  the  beam  is  as  shown 
in  Fig.  9 1 .    These  figures  may  be  understood  by  comparing  them 


Fig.  90. 

with  Figs.  85  and  86.     To  find  the  value  of  the  stress  and  strain 
at  each  point  in  the  bent  beam  proceed  as  follows : 

Consider  the  portion  of  the  beam  which  lies  between  the  radii 
R  and  R  Fig.  88.  This  portion  is  shown  to  a  larger  scale  in 
Fig.  92.  Let  R  be  the  radius  of  curvature  of  the  median  line  //, 
then  the  length  of  pp  is  equal  to  RO  which  is  the  original  length  of 


178 


ELEMENTS   OF   MECHANICS. 


every  filament  of  the  beam  between  the  radii  R  and  R  Fig.  88.  Con- 
sider a  filament  of  the  beam  at  a  distance  y  above  the  median  line, 
y  being  considered  negative  for  filaments  below  the  median  line. 
The  radius  of  curvature  of  this  filament  \?>  R  -\-  y  and  its  length  is 


Fig.   91 


{R  -f  y^O.  Therefore,  the  increase  of  length  of  this  filament  due 
to  the  bending  of  the  beam  is  yQ^  and,  expressing  this  increase 
of  length  as  a  fraction  of  the  original  length  RQ^  we  have 


which  expresses  the  value  of  the  longitudinal  strain  at  any  point 


Fig.  92. 


in  the  bent  beam,  /S  being  positive  where  y  is  positive  and  negative 
where  y  is  negative. 


ELASTICITY. 


179 


The  longitudinal  stress  P  (force  per  unit  of  area,  as  shown  in 
Fig.  91)  at  each  point  of  the  beam  is  equal  to  E^^  according  to 
equation  (48),  where  E  is  the  stretch  modulus  of  the  material  of 
the  beam.     Therefore,  using  y\R  for  y8,  we  have 

_      E 


R 


y 


(") 


The  total  force  action  across  a  complete  section    ab   of  the 
beam  shown  in  Fig.  93,  that  is,  the  total  force  action  of  the  por- 


Fig.  93. 


tion  AA  upon  the  portion  BB^  is  a  torque  about  an  axis  0 
perpendicular  to  the  plane  of  the  figure.  This  axis  is  shown  as 
the  line    00   in  Fig.  94,  which  is  a  sectional  view  of  the  beam, 


\ 


O — 


b  being  the  breadth  of  the  beam  and  d  its  depth, 
action  is  expressed  by  the  equation 


The  torque 


l80  ELEMENTS   OF   MECHANICS. 

^=12-^  ("0 

But  the  torqup  action  across  every  section  of  the  beam  between 
the  two  supports    55   in  Fig.  88  is  equal  to    Wx,    so  that 

^•^=iV-^-  (iv) 

from  which  E  may  be  calculated  when  W^  x,  b,  d  and  R  are 
known. 

Derivation  of  equation  (iii).  — The  portion  of  the  beam  between  the  two  parallel 
lines  Ay,  in  Fig.  94,  has  a  sectional  area  equal  to  b  •  Aj,  and  the  force  action 
across  each  unit  of  area  of  this  portion  of  the  beam  is  EyiR,  according  to  equation 
(ii)  above.  Therefore  the  total  force  action  across  the  area  b  >  d^y  is  Eby  l^y\ Ry 
which,  multiplied  by  the  lever  arm  y,  gives  the  torque  action  about  00  which  is 
due  to  the  portion  of  the  beam  b  •  Ay.     Therefore, 

£b 
AT=-^-y^-Ay 

whence  by  integrating  between  the  limits  j  =  —  d/2  and  y  =  -{-  {i/2,  we  have 
equation  (iii). 

92.  Important   practical  relations  between  longitudinal  stress 

and  strain.*  —  The  important  practical  aspects  of  the  relation  be- 
tween longitudinal  stress  and  strain  may  best  be  brought  out  by 
considering  the  behavior  of  a  steel  rod  (a  test  piece)  which  is 
subjected  to  a  continually  increasing  longitudinal  stress.  Thus, 
the  ordinates  of  the  curves  in  Fig.  95  represent  the  values  of  an 
increasing  longitudinal  stress  in  pounds-weight  per  square  inch, 
and  the  abscissas  represent  the  corresponding  elongations,  in 
hundredths,  of  a  rod  of  ordinary  bridge  steel. 

Up  to  a  point  p,  the  position  of  which  is  not  very  sharply  de- 
fined, the  strain  (elongation  per  unit  initial  length)  is  very  exactly 
proportional  to  the  stress  (stretching  force  per  unit  of  sectional 
area) ;  that  is  the  stress-strain  curve  is  a  straight  line  from  O 
to  /.  Beyond  the  point  /  the  stress-strain  line  is  very  slightly 
curved  until  the  point  g  is  reached  where  the  steel  begins  to  yield 

*The  student  is  referred  to  the  splendid  treatise  on  **  The  Materials  of  Construc- 
tion "  by  J.  B,  Johnson,  Wiley  and  Sons,  1898,  for  a  full  discussion  of  this  subject. 


ELASTICITY. 


18 


very  greatly.  This  yielding  takes  place  rather  irregularly  until 
the  whole  test-piece  has  yielded,  it  alters  the  temper  of  the  steel, 
and  the  steel  then  sustains  an  increased  stress  which  reaches  a 
maximum  at  the  point  /.  The  metal  then  begins  to  be  weakened 
by  the  continued  increase  of  length,  and  finally  the  rod  breaks  at 
the  point  b. 

The  point  /  marks  the  true  elastic  limit ;  the  point  q,  which  is 
sometimes   called  the  yield  pointy   marks  what  is  for  practical 


75  - 

"  ■" 

~ 

~ 

- 

■ 

T 

■«-. 

•^ 

*: 

^ 

^ 

\t 

N 

S 

y 

^ 

J. 

a 

1 

s. 

*^60  - 

/ 

4^ 

\ 

i 

/ 

\ 

(^ 

« 

J 

r 

h 

s?     J 

>  ^ 

f 

y 

^ 

*- 

V 

r\. 

\  •. 

'"" 

- 

Kk 

p 

m'*-5 

y 

0 

g 

/ 

<N 

/ 

0 

/ 

H 

/ 

§ 

/ 

g 

/ 

S 

/ 

•9  rv 

/ 

/ 

i 

/ 

« 

/ 

/ 

i- 

J 

- 

5  10  15  20  25  30 

Stretch,  hundredths  for  curve  A,  ten-thousandths  for  curve  B 

Fig.  95. 


purposes*  called  the  elastic  limit,  and  the  point  t  marks  what  is 
called  the  tensile  strength  of  the  steel.  The  most  important  of 
these  points  is  the  yield  point  or  elastic  limit,  commercially  so 
called,  because  the  steel  breaks  after  a  very  few  applications  of 
a  stress  that  exceeds  the  elastic  limit.  Everyone  is  familiar  with 
this  fact  in  that  a  wire  may  be  easily  broken  by  bending  it  repeat- 
edly beyond  the  elastic  limit,  and  one  can  easily  imagine  how 

*  Because  of  the  difficulty  of  locating  the  point  /  accurately. 


l82 


ELEMENTS   OF   MECHANICS. 


short-lived  a  bridge  or  a  steel  rail  would  be  if  it  were  strained 
beyond  the  elastic  limit  by  every  passing  train.  The  total  elonga- 
tion of  a  sample  of  steel  at  the  breaking  point  b,  Fig.  95,  is  an 
important  indication  of  the  toughness  of  the  steel.  The  follow- 
ing table  gives  the  important  properties  of  several  grades  of  steel. 

TABLE. 

Physical  properties  of  steel. 


Carbon  content 
in  per  cent. 

Elastic  limit  pounds 
per  square  inch. 

Tensile  strength 

pounds  per  square 

inch. 

Elongation  in  4 

inches  in  per 

cent. 

Stretch  modulus  in 

pounds  per  square 

inch. 

0.17 

0.55 
0.82 

51,000 
57,000 
63,000 

68,000 
106,100 
142,250 

33.5 
16.2 

8.5 

29,800,000 

93.  Resilience.  —  The  work  done  per  unit  volume  in  straining 
a  substance  to  its  elastic  limit  is  called  the  resilience  of  the  sub- 
stance. This  work  per  unit  volume  is  equal  to  one  half  the 
product  of  the  limiting  stress  and  the  limiting  strain,  according 
to  equation  (49),  and  it  is  represented  by  the  area  under  the 
straight  portion  of  the  curve  in  Fig.  95.  For  example,  the 
resilience  of  o.  8 2-per-cent. -carbon  steel  is 


pounds 


1X63,000  -...  2      X  0.002  I  =66.  I   . T; 


inch-pounds  foot-pounds 


=  5-5 


inch^ 


that  is,  5.5  foot-pounds  per  cubic  inch.  Thus  it  would  take  100 
cubic  inches  of  this  steel  (about  25  pounds)  made  into  a  spring 
to  store  sufficient  energy  to  supply  one  horse-power  for  one 
second,  provided  the  spring  could  be  so  designed  as  to  be  strained 
in  every  part  to  its  elastic  limit  when  wound  up.  The  resilience 
of  spring  steel  maybe  as  high  as  10  or  12  foot-pounds  per  cubic 
inch,  the  resilience  of  good  cast  iron  is  about  0.5  foot-pound 
per  cubic  inch.  ' 

The  resilience  of  a  substance  is  a  measure  of  its  strength  to 
withstand  a  sudden  shock,  inasmuch  as  a  blow  of  a  hammer,  for 
example,  bends  a  bar  until  the  kinetic  energy  of  the  hammer  is 
all  used  in  bending  the  bar.  A  structure  subject  to  shocks  should 
be  made  of  highly  resilient  material. 


ELASTICITY. 


183 


\ 


94.  Elastic  hysteresis.  —  In  nearly  all  substances  there  is  more 
or  less  of  a  tendency  for  strain  to  persist  after  the  stress  has 
ceased.  This  is  of  course  very  markedly  the  case  where  a  sub- 
stance is  strained  beyond  the  elastic  limit,  but  in  many  substances 
the  elastic  limit  is  by  no  means  sharply  defined,  and  very  slight 
strains  do  not  entirely  disappear  when  the  stress  ceases.  When 
a  substance  is  subjected  to  a  stress  which  increases  and  de- 
creases periodically  between  two  limiting  values  S^  and  S^,  the 
relation     between     stress 

and  strain  is  somewhat  as 

indicated  in  Fig.  96,  where 

ordinates  represent  stress 

and    abscissas     represent 

strain.     The  branch  a  of 

the  curve   represents  the 

relation     between     stress 

and  strain  while  stress  is 

increasing,  and  the  branch 

^  represents  the    relation 

between  stress  and  strain 

while  stress  is  decreasing. 

This     divergence    of  the 

curve  of  stress  and  strain  for  increasing  and  decreasing  stress 

is    called    elastic    hysteresis.       The    increasing    and    decreasing 

stress  is  here  supposed  to  increase  and  decrease  very  slowly. 

If  the    increase    and  decrease    is    rapid    the  divergence  of  the 

two  curves  a  and  b^  Fig.   96,  is  due  to  hysteresis  and  also  to 

elastic  lag. 

95.  Elastic  lag ;  viscosity.  —  Many  substances,  glass  for  ex- 
ample, when  subjected  to  stress,  take  on  a  certain  amount  of 
strain  quickly,  after  which  the  strain  increases  slowly  for  a  time ; 
and  when  the  stress  is  relieved,  a  remnant  of  the  strain  persists 
for  a  time.     This  phenomenon  is  called  elastic  lag. 

The  strain  of  some  substances,  such  as  pitch,  continues  to  in- 
crease indefinitely,  although  it  may  be  very  slowly,  when  they 


Fig,  96. 


1 84 


ELEMENTS   OF   MECHANICS. 


are  under  stress.  Such  substances  are  said  to  be  viscous.  Nearly 
all  metals  are  viscous  when  subjected  to  great  stress. 

Elastic  hysteresis,  elastic  lag,  and  viscosity  cause  energy  to  be 
dissipated  in  a  substance  when  it  is  strained.  Thus  the  vibrations 
even  of  a  steel  spring  die  away  rapidly  in  a  vacuum,  on  account 
of  the  conversion  of  energy  into  heat  as  the  spring  is  repeatedly 
distorted. 

96.  Elastic  fatigue.  —  The  repeated  application  of  a  stress 
weakens  a  metal  so  that  it  will  break  under  less  than  its  normal 
breaking  stress,  or  less  even  than  the  stress  corresponding  to  its 


^     70 


n 

.lis 


1         2         3         4         5         6         7  S 


60 


III'' 


repetitions  of  stress  in  millions 
Fig.  97. 

elastic  limit.     Fig.  97  shows  the  decease  in  tensile  strength  of  a 
sample  of  mild  steel  with  repetitions  of  the  stress. 

Continued  repetition  of  stress  causes  an  increase  in  the  amount 
of  energy  dissipated  by  elastic  lag  and  viscosity.  Thus  the  vibra- 
tions of  a  torsion  pendulum  die  away  faster  after  it  has  been  kept 
vibrating  for  several  days,  than  at  first. 


HYDROSTATIC  PRESSURE  AND  ISOTROPIC  STRAIN. 
97.  Hydrostatic  pressure.* — A  fluid  at  rest  not  only  pushes 
normally  against  a  surface  which  is  exposed  to  its  action,  but  two 
contiguous  portions  of  a  fluid  at  rest  always  push  on  each  other 
at  right  angles  to  a  small  plane  q  which  may  be  imagined  to 
separate  them  as  indicated  in  Fig.  98.  Whatever  the  direction 
of  the  small  plane  q  may  be,  \}i\Q.  force  action  per  unit  area  across 

*  Hydrostatic  pressure  is  discussed  also  in  the  chapter  on  hydrostatics. 


k 


ELASTICITY.  1 85 

it  is  the  same.  This  fact  was  first  pointed  out  by  Pascal  (1623- 
1662)  and  it  is  sometimes  called  Pascal's  principle.*  The  force 
action  per  unit  area  at  a  point  in  a  fluid  is  generally  represented  by 
the  letter  p  and  it  is  called  the  hydrostatic  pressure  at  the  point. 

98.  Isotropic  strain.  —  When  a  substance  like  glass  or  cast 
metal  is  subjected  to  an  increase  of  hydrostatic  pressure  the  sub- 
stance is  reduced    in    size  without 

being  changed  in  shape ;  such  a 
strain  is  called  an  isotropic  strain. 
Let  V  be  the  original  volume  of  the 
substance,  and  let  v  be  the  diminu- 
tion of  volume  due  to  the  increase 
of  pressure.  It  is  convenient  to  irj  T  L^^/ 
express  7/  as  a  fraction  of  V^  and  It^-  ~-  -  /  -  p-^ 
this  fraction  vj  V  is  used  as  a  meas- 
ure of  the  isotropic  strain.  .--^^. - .  , .  „  .^  _-^ ^^^,-^-^^ 

99.  Bulk  modulus  of  a  substance.  p.    ^g 
—  The    diminution    of   volume    of 

a  substance  per  unit  of  original  volume  {vj  V)  is  proportional 
to  the  increase  of  hydrostatic  pressure,  except,  of  course,  when 
the  increase  of  pressure  is  very  great  and  the  decrease  of  volume 
considerable.  Therefore,  for  small  changes  of  volume,  the  ratio 
of  increase  of  hydrostatic  pressure  to  decrease  of  volume  per 
unit  volume  is  a  constant  for  a  given  substance  (at  a  given  tem- 
perature).    That  is 

V 

in  which  V  is  the  volume  of  a  substance  at  a  given  pressure,  v  is 
the  decrease  of  volume  due  to  an  increase  of  pressure  /,  and  B 
is  a  constant  for  the  given  substance.  This  quantity  B  is  called 
the  bulk  modulus  of  the  substance.     The  reciprocal  of  B  is  called 

*  The  student  should  attempt  to  establish  this  principle  by  considering  the  equi- 
librium of  a  prism  shaped  portion  of  a  fluid  at  rest. 


l86    ■  ELEMENTS   OF   MECHANICS. 

the   coefficient   of  co^npressibility  of  the   substance.     Therefore, 
writing  C  for  ijB  equation  {^\)a  becomes 

or 

v=pVC  (5i> 

The  coefficient  of  compressibihty  of  a  substance  is  the  change 
of  volume  per  unit  original  volume  per  unit  increase  of  pressure, 
and,  according  to  equation  (51^,  the  decrease  of  volume  of  a 
substance  due  to  a  given  increase  of  pressure  is  equal  to  the 
product  of  the  increase  of  pressure,  the  original  volume,  and 
the  coefficient  of  compressibility. 

TABLE.* 

Coefficients  of  compressibility  at  20°  C.  for  moderate  increase  of  pressure. 

( Decrease  of  volume  per  unit  volume  per  atmosphere  increase  of  pressure. ) 


Substance. 

C  X  io«. 

Ether 

170.0 

Alcohol 

lOI.O 

Water 

46.0 

Glass 

2.2 

Steel 

0.68 

100.  Potential  energy  of  isotropic  strain. — The  potential  energy 
per  unit  volume  of  an  elastic  substance  under  increased  pressure 
is  equal  to  one  half  the  product  of  the  increase  of  hydrostatic 
pressure  /  and  the  strain  vj  V,  or  it  is  equal  to  one  half  the 
product  of  the  bulk  modulus  B  and  the  square  of  the  strain 
{v^l  V^).  This  relation  may  be  derived  in  a  manner  very  similar 
to  the  proof  of  equations  (49)  and  (50). 

101.  Compressibility  of  gases.  Boyle's  law.  —  Solids  and 
liquids  generally  decrease  but  sHghtly  in  volume  when  subjected 
to  increase  of  pressure.  Thus  the  volume  of  water  decreases 
about  one  ten-thousandth  part  when  subjected  to  an  increase  of 

*See  Pkysikalish-Che?nische  Tabellen  by  Landolt  and  Bornstein,  Berlin,  1895, 
for  a  very  complete  collection  of  data  of  all  kinds. 


i 


ELASTICITY.  1 8/ 

pressure  oi  30  pounds  per  square  inch,  and  the  volume  of  steel 
decreases  about  one  ten-thousandth  part  when  subjected  to  an 
increase  of  pressure  of  2,000  pounds  per  square  inch. 

Gases,  on  the  other  hand,  decrease  greatly  in  volume  when 
subjected  to  increase  of  pressure.  The  remarkable  contrast  be- 
tween water  and  air  in  regard  to  compressibility  may  be  shown 
by  filling  a  bicycle  pump  with  air  and  then  with  water,  and  strik- 
ing the  piston  rod  in  each  case  with  a  hammer.  The  air  will  be 
found  to  act  as  a  cushion  and  the  water  will  appear  to  be  as  solid 
as  if  the  whole  pump  barrel  and  piston  were  one  piece  of  steel. 
When  a  steam  engine  is  started,  the  water  which  usually  collects 
in  the  steam  pipes  may  enter  the  cylinder  in  sufficient  quantity  to 
cause  the  moving  piston  to  burst  the  cylinder  head. 

When  the  temperature  of  a  gas  is  kept  at  a  constant  value ^  the 
volume  of  the  gas  is  inversely  proportional  to  the  pressure  to  which 
the  gas  is  subjected.     That  is 

k 

P 
or 

pv^k  (52) 

in  which  v  is  the  volume  of  a  given  amount  of  gas,  /  is  the  pres- 
sure of  the  gas,  and  /^  is  a  constant.  This  relation,  which  is 
known  as  Boyle's  law,  was  discovered  by  Robert  Boyle\*  in 
1 660,  and  more  completely  established  by  Mariotte,  who  discov- 
ered it  independently  in  1676.  It  is  very  accurately  true  of  such 
gases  as  hydrogen,  nitrogen  and  oxygen  at  ordinary  temperatures 
and  pressures,  but  all  gases  deviate  from  it  appreciably,  especially 
at  low  temperatures  and  under  great  pressures.  See  the  discus- 
sion of  the  properties  of  gases  in  the  chapters  on  heat. 

SHEARING   STRESS   AND   SHEARING    STRAIN. 
102.  Shearing  stress. — The  type  of  stress  and  strain  in  a  twisted  rod  is  called 
shearing  stress  and  shearing  strain,  and  the  discussion  of  this  type  of  stress  and  strain 
is  somewhat  obscured  by  the  fact  that  there  is  no  familiar  example  in  which  homo- 
geneous shearing  stress  and  shearing  strain  occur  ;  the  stress  and  strain  in  a  twisted 

'^  New  Experiments  touching  the  Spring  of  Air,  Oxford,  1660. 


i88 


ELEMENTS   OF   MECHANICS. 


rod  are  non-homogeneous.  Any  intelligent  discussion  of  shearing  stress  and  shearing 
strain  must,  however,  be  based  on  a  case  in  which  the  stress  and  strain  are  homo- 
geneous. Consider,  therefore,  a  cubical  portion  of  a  substance  ABCDy  Fig.  99,  and 
suppose  that  outward  forces  (5  units  of  force  per  unit  of  area)  act  upon  the  faces  AB 
and  6'Z>,  that  inward  forces  {S  units  of  force  per  unit  of  area)  act  upon  the  faces 
AC  and  BD^  and  that  no  force  at  all  acts  on  the  two  faces  of  the  cube  which  are 
parallel  to  the  plane  of  the  paper.  Then  the  material  of  the  cube  will  be  subject  to 
what  is  called  a  shearing  stress,  and  the  stress  will  be  homogeneous.     The  character 


II  If  ft 


t      t      T      I     Y      Y 


Fig.  99. 

of  the  force  action  between  contiguous  parts  of  the  material  of  the  cube  is  as  follows  : 
A  pull  of  S  units  of  force  per  unit  area  acts  across  any  plane  q^  which  is  parallel  to 
the  faces  AB  and  CD  of  the  cube  ;  a  ptisk  of  S  units  of  force  per  unit  area  acts 
across  any  plane  q^  which  is  parallel  to  the  faces  AC  and  BD  of  the  cube  ;  a  slid- 
ing force,  or  tangential  force,  as  it  is  called,  of  S  units  of  force  per  unit  of  area  acts 
across  any  plane  ^3  or  ^^  which  is  parallel  to  the  diagonal  plane  AD  or  BC\  and 
no  force  at  all  acts  across  a  plane  which  is  parallel  to  the  plane  of  the  paper. 

To  show  that  the  force  action  across  the  diagonal  planes  q^  and  q^  is  a  tangential 
force  action  and  that  the  tangential  or  sliding  force  is  S  units  of  force  per  unit  area, 
consider  any  unit  cube  abed  of  the  material.  The  area  of  each  face  of  this  cube  is 
unity,  and  the  area  of  the  diagonal  plane  be  is  v/2  units.  The  total  force  acting  on 
the  face  bd  is  a  push  of  S  units,  the  total  force  acting  on  the  face  ed  is  a  pull  of  S 
units,  and  the  resultant  of  these  two  forces  is  a  force  parallel  to  be  and  equal  to 
\/2S  as  shown  in  Fig.  100.  Similarly,  the  resultant  of  the  forces  acting  on  the 
faces  ab  and  a^r  is  a  force  parallel  to  eb  and  equal  to  V^S.  Therefore,  the  force 
action  across  the  diagonal  plane  be  is  a  tangential  force  action  equal  to  V^S,  which, 
divided  by  the  area  of  the  diagonal  plane  be,  gives  a  tangential  force  action  of  6"  units 
of  force  per  unit  of  area. 


ELASTICITY. 


189 


It  can  be  shown  in  the  same  way  that  the  force  action  across  q^^  Fig.  99,  is  a  tan- 
gential force  of  S  units  per  unit  of  area.  It  is  on  account  of  the  purely  tangential 
forces  across  q^  and  q^  that  this  type  of  stress  is  called  a  shearing  stress. 


Fig.  100. 

103.  Shearing  strain. — The   effect  of  the  prescribed   stress  in   Fig.  99  is  to 
shorten  the  cube  in  the  direction  of  the  push  and  to  lengthen  the  cube  by  an  equal 


in 

A                             1 

\  / 

I 


Fig.  101. 


190 


ELEMENTS   OF   MECHANICS. 


amount  in  the  direction  of  the  pull,  without  changing  the  dimensions  of  the  cube  in  a 
direction  at  right  angles  to  the  plane  of  the  paper  in  Fig.  99.  This  distortion  is 
represented  by  the  dotted  rectangle  in  Fig.  loi,  and  the  dotted  rhombus  mnop  in 
Fig.  loi  is  a  figure  which  was  square  in  the  unstrained  material.  The  angle  Q  is  less 
than  90°  and  the  value  of  the  angle  90° ^6^  (expressed  as  a  fraction  of  a  radian)  is 
called  the  angle  of  the  shearing  stj-am,  or  simply  the  angle  of  shear.  It  is  usually  repre- 
sented by  the  letter  (p,  and  it  is  used  as  a  measure  of  the  shearing  strain. 

Let  L  be  the  original  length  of  each  edge  of  the  cube  in  Fig.  99,  and  let  /  be  the 
increase  of  length  in  the  direction  of  the  pull  and  the  decrease  of  length  in  the  direc- 
tion of  the  push.  It  is  convenient  to  express  /  as  a  fraction  of  L  and  we  will  repre- 
sent this  fraction  by  the  letter  a  (==  Ij L).  It  is  important  to  know  that  the  angle  of 
shear  ^  as  above  defined  is  equal  to    2a.     That  is 


0  — 2a 


(53) 


The  full-line  square  in  Fig.  102  is  the  figure  which  when  distorted  becomes  the 
rhombus  in  Fig.  loi,  and  the  small  triangle  in  Fig.  102  is  enlarged  in  Fig.  103.     The 


Fig.  102. 


Fig.  103. 


angle   ^/2   is  very  small  and  it  is  therefore  sensibly  equal  to  gh   divided  by  gn. 
That  is,    ^/2    is  equal  to   //Z.   (^a)    so  that   ^  ^  2a. 

104.  Slide  modulus  of  a  substance.  —  The  angle  of  shear  ^  produced  in  Fig. 
99  by  the  shearing  stress  S  is  proportional  to  S,  according  to  Hooke's  law,  so  that 
the  ratio  SJ^  is  a  constant  for  a  given  substance,  within  the  limits  of  elasticity.  This 
constant  is  called  the  slide  modulus  of  the  substance  and  it  is  represented  by  the  letter 
n.     Therefore  we  have 

«  =  l  (54) 


The  slide  modulus  of  a  substance  is  sometimes  called  the  shearing  modulus  of  the 


ELASTICITY. 


191 


substance.     It  is  approximately  equal  to  |  of  the  stretch  modulus  (Young's  modulus) 
for  metals. 

105.  Discussion  of  a  twisted  rod.  —  Consider  a  cylindrical  rod  of  radius  R  and 
length  L,  and  suppose  that  one  end  of  the  rod  is  fixed  while  the  other  is  turned 
through  the  angle  d  so  as  to  twist  the  rod.  Consider  a  cylindrical  shell  of  the  material 
of  the  rod  of  which  the  radius  is  y,  and  imagine  this  cylindrical  shell  to  be  cut  along 
one  side  and  laid  out  flat  so  that  it  may  be  pictured  on  a  flat  surface.^     Figures  104 

-5}rr-^- >|  ^^ ; zTfY — ^ — 


u 


I 
I 

I 
f 
1 
I 
I 
I 

I 
I- 

I 

z 

I 
I 
I 
I 
I 
I 
I 
\ 
I 


If 


i 


I 


\ 


/% 


B^i 


v 


\ 


Fig.  104. 


Fig.  105. 


and  105  represent  the  cylindrical  shell  laid  out  flat,  or  developed,  in  this  way  ;  Fig. 
104  before  twisting,  and  Fig.  105  after  twisting.  The  line  AB  which  is  parallel  to 
the  axis  of  the  rod  in  the  untwisted  rod  takes  the  position  A^ B^  after  the  rod  is 
twisted,  and  the  small  square  becomes  a  rhombus.  The  portions  of  the  rod  in  the 
cylindrical  shell  under  consideration  are  subjected  to  a  shearing  strain  of  which  the 
angle  of  shear  is 

<i>  =  T  (55) 


*  This  discussion  of  the  strain  in  a  twisted  rod  may  be  made  more  easily  intelligible 
by  means  of  a  model  as  follows  :  A  tin  cylinder  about  20  cm.  in  diameter  and  35  cm. 
long,  has  wooden  disks  fixed  in  each  end.  On  one  end  is  an  additional  wooden  disk 
which  turns  on  a  nail  which  is  in  the  axis  of  the  cylinder.  Loosely  woven  muslin  is 
tacked  at  one  end  to  the  movable  disk  and  at  the  other  end  to  the  fixed  disk.  This 
muslin  fits  the  tin  cylinder  closely,  and  the  seam  at  one  side  is  sewed.  On  this  muslin 
a  small  square  may  be  drawn  like  Fig.  104,  and  also  small  circles,  and  when  the 
movable  disk  is  turned  through  a  considerable  angle,  the  distortion  of  the  square  and 
of  the  circles  will  give  a  clear  idea  of  the  character  of  the  strain  in  a  cylindrical  shell 
of  a  twisted  rod. 


192  ELEMENTS   OF   MECHANICS. 

as  is  evident  from  Fig.  105.  The  character  of  the  force  action  between  contiguous 
portions  of  the  material  of  the  rod  may  be  understood  by  comparing  Fig.  105  with  Fig. 
99,  There  is  a  tangential  force  action  across  every  vertical  plane  q^  and  across  every 
horizontal  plane  ^^,*  there  is  a  normal  pull  across  every  plane  like  ^j,  and  a  normal 
push  across  every  plane  like  q^  ;  and  the  force  per  unit  area,  Sy  in  each  case  is  equa^ 
to  «^,  according  to  equation  (54).  Therefore,  substituting  for  ^  its  value  from  equa- 
tion (55),  we  have 

^  =  ?.  .     (56) 

The  tangential  stress  across  vertical  planes  like  q^.  Fig.  105,  is  concentrated  at  the 
bottom  of  a  sharp  groove  cut  in  the  rod  parallel  to  its  axis,  like  a  key  seat  in  a  shaft, 
and  such  a  groove  therefore  weakens  the  rod  very  much  indeed. 

Constant  of  torsion  of  a  rod  or  wire.  —  The  total  force  action  across  a  complete 
section  of  a  twisted  rod  is  a  torque  /'about  the  axis  of  the  rod  and  the  value  of  the 
torque  is 

T= ^T-  (57) 

in  which  n  is  the  slide  modulus  of  the  material  of  the  rod  or  wire,  R  is  the  radius  of 
the  rod  or  wire,  L  is  the  length  of  the  rod  or  wire,  and  B  is  the  angle  through  which 

one  end  of  the  rod  or  wire  is  twisted.  The  negative 
sign  is  written  for  the  reason  that  the  torque  tends 
to  reduce  Q,-\  that  is,  7" and  B  are  opposite  in  sign. 
This  equation  (57)  shows  that  T  is  proportional  to 
By  and  the  proportionality  factor  TTnR*0J2L  is 
called  the  constant  of  torsion  of  the  wire  or  rod. 
If  the  constant  of  torsion  of  a  rod  or  wire  is  deter- 
mined by  observing  the  angle  of  twist  fj  produced 
by  a  known  torque,  the  slide  modulus  of  the  material 
may  be  calculated  from  equation  (57). 

Proof  of  equation  {jy). — Let    Fig.   106  rep- 
resent a  sectional  view  of  the  rod.      Consider  the 
Fig.  106.  narrow  annulus  of  width  Ar  and  radius  r,  as  shown 

by  the  dotted  lines.  The  force  action  per  unit  area 
across  this  annulus  is  rnB / L  according  to  equation  (56),  and  this  force  action  is  at 
right  angles  to  r  at  each  point  of  the  annulus.     Therefore  the  torque  action,  about  the 

*  The  force  actions  between  contiguous  portions  of  a  twisted  rod  are  inferred  from 
the  character  of  the  distortion  at  each  point  as  represented  in  Fig.  105.  These  force 
actions  may  be  made  clearly  evident  by  the  use  of  two  models,  as  follows:  [a)  A 
bundle  of  smooth  squaie  pine  sticks  bound  together  and,  if  desired,  turned  to  a 
cylindrical  shape,  shows  sliding  along  vertical  planes  like  q^,  Fig.  105,  when  the 
bundle  of  sticks  is  twisted.  (3)  A  brass  tube  with  a  slit  along  a  helical  line  which  is 
at  each  point  inclined  at  an  angle  of  45°  to  the  axis  of  the  tube,  is  as  strong  as  an 
uncut  tube  to  withstand  a  twist  in  one  direction  (the  faces  of  the  slit  push  against  each 
other  normally),  but  the  slit  opens  when  the  tube  is  twisted  in  the  other  direction. 

f  The  reacting  torque  of  the  twisted  rod  is  here  referred  to. 


ELASTICITY.  193 

axis  of  the  rod,  of  the  force  which  acts  across  unit  area  of  the  annulus  is  ^X  ^«^/^> 
which,  multiplied  by  the  area  zirr  •  Ar  of  the  annulus,  gives  the  total  torque  action 
A  T  across  the  annulus.     That  is 

.  ^      zvrrhtd    . 
Ar=^^r— .Ar 

or 


L    Jo 


2.L 


GENERAL   EQUATIONS   OF   STRESS   AND   STRAIN. 

106.  Principle  stretches  of  a  strain.  — A  small  spherical  portion  of  a  body  always 
becomes  an  ellipsoid  when  the  body  is  distorted.  A  distortion  which  changes  a  sphere 
into  an  ellipsoid  consists  always  of  simple  increase  or  decrease  of  linear  dimensions  in 
three  mutually  perpendicular  directions.  These  mutually  perpendicular  directions  are 
called  the  axes  of  the  strain,  and  the  increase  of  length  per  unit  length  {IJ  L)  in  the 
directions  of  the  respective  axes  are  called  the  principal  stretches  of  the  strain.  The 
principal  stretches  of  a  strain  are  represented  by  the  letters  f  g,  and  h. 

107.  Principal  pulls  of  a  stress.  — Imagine  a  small  plane  area  q,  called  a  section, 
in  the  interior  of  a  body  under  stress.  The  portions  of  the  body  on  the  two  sides  of 
this  section  exert  on  each  other  a  definite  force  in  a  definite  direction,  and  the  force  per 
unit  area  of  the  section  is  called  the  stress  on  the  section.  When  the  force  is  norma] 
to  the  section,  the  stress  on  the  section  is  called  2^.  pull,  positive  or  negative  as  the  case 
may  be.  When  the  force  is  parallel  to  the  section,  the  stress  on  the  section  is  called 
a  tangential  stress. 

The  conditions  of  equilibrium  of  a  small  portion  of  a  body  under  stress  require  * 
that  at  a  point  in  the  body  there  be  three  mutually  perpendicular  sections  across 
which  the  force  action  is  normal,  or  on  which  the  stress  is  a  pull.  These  three  sec- 
tions are  called  the  principal  sections  of  the  stress,  the  three  lines  perpendicular  to 
them  are  called  the  axes  of  the  stress,  and  the  pulls  on  the  three  sections  are  called 
the  principal  pulls  of  the  stress.  These  three  pulls  constitute  the  stress  at  the  point 
and  if  these  pulls  are  specified  the  stress  is  completely  determined. 

108.  General  equations  of  stress  and  strain.  —  Let  7^  be  a  longitudinal  stress, 
that  is,  a  simple  pull,  in  the  direction  of  the  x-axis  of  reference.  Such  a  stress  causes 
a  stretch  aF  in  the  direction  of  the  jf-axis,  and  a  negative  stretch  — bF  in  the 
directions  of  the  y  and  z-axes.  Therefore,  writing  f\  g^  and  h^  for  the  three 
stretches  due  to  the  simple  pull  F^  we  have 

f'^aF 

g^  =  -bF  (i) 

hf  =  —  bF 

Similarly,  let  6^  be  a  longitudinal  stress  in  the  direction  of  the  ^-axis  of  reference, 
and  let  f'\  g'^,  and  h^^  be  the  three  stretches,  parallel  to  x,  y,  and  z  axes  respectively, 
produced  by  G.     Then  we  have  : 

f^^  =  —  bG 

g''  =  aG  '  (ii) 

/^//  =  _36; 
*  See  Elasticity,  theory  of  in  Encyclopedia  Britannica,  9th  Exi. 
13 


194  ELEMENTS   OF   MECHANICS. 

Similarly,  let  J/  he  a.  longitudinal  stress  parallel  to  the  2-axis  of  reference,  and  let 
/^^^*  g'^'i  and  h'^'  be  the  three  stretches,  parallel  to  the  jc,  ^,  and  z  axes,  respec- 
tively, produced  by  H.     Then  we  have  : 

ff'^=:^  —  bH 

g^ff  =  —bH  (iii) 

hf^'  =  aH 

Experiment  shows*  that  the  stretch  produced  in  any  direction  by  a  number  of  pulls 
acting  together  is  equal  to  the  sum  of  the  stretches  in  that  direction  produced  by  the 
respective  pulls  acting  separately.     Therefore  : 

g=g'-Vg''^g'"  (iv) 

in  which  /,  g,  and  h  are  the  stretches,  parallel  to  the  x^  y,  and  z  axes,  respectively, 
produced  by  the  three  pulls  F,  G,  and  If  acting  together.  Therefore  substituting  the 
values  of /^,  /^^,  /^^^,  g^,  g'^,  g'^^y  h\  h^^,  and  h'^'  from  equations  (i),  (ii),  and 
(iii)  in  equation  (iv),  and  we  have  : 

f=aF—bG  —  bH 

g  =  —  bF-\-aG  —  bH  (58) 

h  =  —  bF—bG^aH 

These  equations  give  the  strain  (/,  g^  and  k)  which  is  produced  in  an  isotropic 
elastic  solid  by  any  stress  (7%  G",  and  H)^  and  it  shows  that  an  isotropic  elastic  solid 
has  but  two  constants  of  elasticity  a  and  b.  In  fact  the  quantity  a  is  equal  to  \\E 
and  the  quantity  b  is  equal  to  aa^  where  a  is  the  value,  ^''//3,  of  Poisson's  ratio. 
See  Art.  87.  Starting  with  these  relations,  it  is  very  easy  to  derive  expressions  for 
the  bulk  modulus  B  and  for  the  slide  modulus  n  of  a  substance  in  terms  of  F  and  g 
by  using  equations  (58)  ;  considering  that  7^=  G  =  I/=J>  in  the  case  of  hydro- 
static pressure,  and  that  F^=-\-S,  G  =:  —  S,  and  I/=o  in  the  case  of  a  shear- 
ing stress. 

Problems. 
109.  A  helical  spring  is  elongated  by  an  amount  of  1.2  inches 
when  a  4-pound  weight  is  hung  upon  it.      How  much  additional 
elongation  is  produced  by  i  pound  additional  weight  ?     By  two 

*  In  general,  any  effect  which  is  proportiottal  to  a  cause,  may  be  resolved  into  parts 
which  correspond  to  the  parts  of  the  cause.  Thus  a  spring  stretches  in  proportion  to 
the  stretching  force.  One  kilogram  produces,  say,  one  centimeter  elongation  ;  two 
kilograms  produce  two  centimeters  elongation,  which  is  one  centimeter  for  each  kilo- 
gram.    See  footnote  to  Art.  32. 


ELASTICITY.  195 

pounds  additional  weight  ?     By  three  pounds  additional  weight  ? 
By  four  pounds  additional  weight  ? 

Note. — Assume  in  this  problem,  and  in  those  that  follow,  that  the  elastic  limit  is 
not  exceeded. 

110.  The  middle  of  a  long  beam  is  depressed  2  inches  by  a  load 
of  5,000  pounds.  How  much  will  it  be  depressed  by  a  load  of 
15,000  pounds? 

111.  A  long  rod  is  fixed  at  one.  end,  and  a  twisting  force,  or 
torque,  of  100  pound-inches  applied  at  the  free  end  causes  the  free 
end  to  turn  through  an  angle  of  10^.  What  torque  would  be 
required  to  turn  the  free  end  of  the  rod  through  26°  ? 

112.  A  rod  2  inches  in  diameter  and  20  feet  long  is  stretched 
to  a  length  of  20  feet  and  ^  inch  by  a  force  of  10,000  pounds- 
weight.  What  is  the  value  of  the  longitudinal  stress,  and  what  is 
the  value  of  the  longitudinal  strain  ? 

113.  A  rod  20  feet  long  and  i  inch  in  diameter  is  subjected  to 
a  pull  of  20,000  pounds  per  unit  of  sectional  area  causing  it  to  be 
lengthened  to  20.02  feet,  that  is  one  part  in  a  thousand,  and  caus- 
ing it  to  contract  to  a  diameter  of  0.997  inch,  that  is,  three  parts 
in  ten  thousand.  What  is  the  length  and  what  is  the  diameter 
of  the  rod  when  it  is  subjected  to  a  pull  of  40,000  pounds  per 
unit  sectional  area  ? 

114.  A  wire  200  inches  long  and  o.  i  inch  in  diameter  is  pulled 
with  a  force  of  i  50  pounds.  The  elongation  produced  is  \  inch. 
What  is  the  value  of  the  stretch  modulus  of  the  material  ? 

115.  A  wire  five  feet  long  and  0.06  square  inch  sectional  area 
is  subjected  to  a  stretching  force  of  300  pounds.  The  stretch 
modulus  of  the  material  is  28,000,000  pounds  per  square  inch. 
What  elongation  is  produced  ? 

116.  A  steel  beam  is  bent  so  that  its  middle  line  forms  the  arc 
of  a  circle  600  inches  in  radius.  What  is  the  elongation  per  unit 
length  of  a  filament  2  inches  from  the  middle  line  ? 

117.  The  stretch  modulus  of  the  steel  of  which  the  beam  of  the 
previous  problem  is  made  is  30,000,000  pounds  per  square  inch. 


196  ELEMENTS   OF   MECHANICS. 

What  is  the  pull  (force  per  unit  area  of  course)  of  a  filament  of 
the  beam  2  inches  from  the  middle  line  of  the  beam  ? 

118.  What  is  the  resilience  of  spring  steel  of  which  the  elastic 
limit  is  70,000  pounds  per  square  inch  and  of  which  the  stretch 
modulus  is  30,000,000  pounds  per  square  inch  ? 

119.  A  cork  i  inch  in  diameter  is  pushed  with  a  force  of  20 
pounds-weight  into  a  bottle  which  is  completely  filled  with  water. 
What  hydrostatic  pressure  is  produced  in  the  bottle  ?  Neglect 
the  friction  of  the  cork  against  the  glass  neck  of  the  bottle. 

120.  A  body  subjected  to  hydrostatic  pressure  is  decreased  in 
length,  in  breadth,  and  in  thickness  by  5  parts  in  a  thousand 
(initial).  By  how  many  parts  per  thousand  (initial)  is  the  volume 
reduced  ? 

121.  2,000  cubic  inches  of  water  are  reduced  to  1,880  cubic 
inches  by  an  hydrostatic  pressure  of  3,000  lbs.  per  square  inch. 
What  is  the  value  of  the  bulk  modulus  of  water  ? 

122.  Calculate  the  compressibility  of  water  from  the  answer  to 
the  previous  problem  and  explain  its  meaning. 

123.  A  bicycle  pump  is  full  of  air  at  1 5  pounds  per  square 
inch,  length  of  stroke  is  1 2  inches  ;  at  what  part  of  the  stroke  does 
air  begin  to  enter  the  tire  at  40  pounds  per  square  inch  above  at- 
mospheric pressure  ?  Assume  the  compression  to  take  place  with- 
out rise  of  temperature. 

124.  The  clearance  space  behind  the  piston  of  an  air  compressor 
when  the  piston  is  at  the  end  of  its  stroke  is  ^^^  of  the  volume 
swept  by  pistojt  during  the  stroke.  What  is  the  greatest  pressure 
that  can  be  produced  in  a  compressed  air  reservoir  by  this  com- 
pressor, the  compression  of  the  air  in  the  cyhnder  being  assumed 
to  be  without  change  of  temperature  ? 

Note.  —As  a  matter  of  fact  the  air  in  an  air  compressor  is  heated  very  considerably 
by  the  compression. 

125.  The  piston  of  an  air  pump  is  0.0 1  inch  from  the  bottom 
of  the  cylinder  when  it  is  at  the  end  of  its  stroke,  and  the  pressure 
of  the  air  in  the  clearance  space  is  then  at  atmospheric  pressure. 


ELASTICITY.  1 97 

The  length  of  stroke  is  6  inches.     What  is  the  lowest  vacuum 
which  can  be  produced  by  the  pump  ? 

126.  A  cubical  piece  of  steel  is  shortened  two  parts  in  a  thousand  in  one  direction, 
lengthened  two  parts  in  a  thousand  in  a  direction  at  right  angles  to  the  first,  and 
unchanged  in  dimension  in  the  third  direction,  as  represented  in  Fig.  loi.  "What  is 
the  value  of  the  angle  of  shear  in  degrees  ? 

127.  A  steel  rod  120  inches  long  is  fixed  at  one  end  and  the  other  end  is  turned 
through  5  degrees  of  angle.  Consider  a  small  portion  /  of  the  rod  at  a  distance  of  I 
inch  from  the  axis  of  the  rod.  Find  the  angle  of  shear  <j)  of  this  small  portion  /  of 
the  metal. 

128.  The  slide  modulus  of  the  steel  used  in  the  rod  of  problem  127  is  12  million 
pounds  per  square  inch.     Find  the  shearing  stress  in  the  small  portion  /  of  the  rod. 

129.  A  steel  shaft  500  inches  long  and  3  inches  in  diameter  transmitting  100  horse- 
power is  subject  to  a  torque  of  23,100  pound-inches  of  torque.  The  slide  modulus 
of  the  material  of  the  shaft  is  12,000,000  pounds  per  square  inch.  Calculate  the 
angle  through  which  one  end  of  the  shaft  is  twisted  relative  to  the  other  end. 

130.  The  three  stretches  of  a  strain  are  -|-  0.015,  -\-  0.025  and  — 0.025.  What 
are  the  semi-axes  of  the  ellipsoid  into  which  a  sphere  10  inches  in  radius  is  distorted 
by  this  strain  ?     Strain  supposed  to  be  homogeneous. 

131.  A  force  of  250  pounds  acts  across  a  section  of  which  the  area  is  ^  square  inch. 
What  is  the  value  of  the  stress  on  the  section  ? 

132.  A  square  rod  2X1^  inches  is  subjected  to  a  tension  of  75,000  pounds. 
What  kind  of  stress  acts  across  a  section  of  the  rod  and  what  is  its  value  ? 

133.  Two  long  strips  of  metal  are  lapped  and  fastened  by  a  single  rivet  of  which 
the  sectional  area  is  two  square  inches.  The  two  strips  are  subjected  to  a  tension  of 
10,000  pounds.  What  kind  of  stress  acts  across  the  middle  section  of  the  rivet  and 
what  is  the  value  of  the  stress  ? 

134.  Derive  the  equation  expressing  the  bulk  modulus  of  a  substance  in  terms  of 
its  stretch  modulus  and  Poisson's  ratio.  The  stretch  modulus  of  steel  is  30  million 
pounds  per  square  inch,  and  Poisson's  ratio  for  steel  is  0,28.  Find  the  value  of  the 
bulk  modulus,  and  find  the  value  of  the  coefficient  of  compressibility  and  compare  it 
with  the  value  given  in  the  table  in  Art.  99.  One  atmosphere  is  equal  to  14.7  pounds 
per  square  inch. 

135.  Derive  the  equation  expressing  the  slide  modulus  of  a  substance  in  terms  of 
its  stretch  modulus  and  Poisson's  ratio,  and  calculate  the  slide  modulus  of  steel  using 
the  data  given  in  problem  134. 


CHAPTER   IX. 

HYDROSTATICS. 

109.  Pressure  at  a  point  in  a  fluid.*  —  The  force  with  which  a 
fluid  at  rest  pushes  against  an  element  of  an  exposed  surface  is 
at  right  angles  to  the  element  and  proportional  to  the  area  of  the 
element.  The  force  per  unit  area  is  called  the  hydrostatic  pres- 
sure or  simply  the  pressure  of  the  fluid  at  the  place  where  the 
element  of  area  is  located  and  it  is  usually  represented  by  the 
letter  /.  When  the  pressure  has  the  same  value  throughout  a 
fluid  the  pressure  is  said  to  be  uniform,  when  the  pressure  varies 
from  point  to  point  in  a  fluid  the  pressure  is  said  to  be  non-uni- 
form. When  the  pressure  in  a  fluid  is  uniform  the  total  force  F 
acting  on  an  exposed  plane  surface  is 

F^pa  (59) 

where  a  is  the  area  of  the  surface. 

Examples,  (a)  Steam  presstire.  —  The  piston  of  a  steam  en- 
gine is  pushed  by  a  force  equal  to  pa,  where  a  is  the  area  of 
the  piston,  and  /  is  the  pressure  of  the  steam  in  the  cylinder. 
Every  part  of  the  inside  surface  of  a  steam  boiler  is  pushed  out- 
wards by  the  steam. 

(b)  Atmospheric  pressure.  —  The  force  with  which  the  air 
pushes  on  the  surfaces  of  bodies  does  not  ordinarily  appeal  to  our 
senses.  It  is  shown  however  by  the  collapse  of  a  thin-walled 
vessel  when  the  inside  pressure  is  reduced  by  pumping  out  the 
air.  Atmospheric  pressure  is  also  strikingly  sh  wn  by  means  of 
the  apparatus  known  as  the  Magdeburg  Hemispheres.  This  con- 
sists of  two  metal  cups  which  fit  together  air  tight  and  form  a 
hollow  vessel  from  which  the  air  may  be  removed  by  pumping. 

*  See  Art.  97. 


HYDROSTATICS. 


199 


The  pressure  of  the  outside  air  then  holds  the  cups  together  and 
a  considerable  effort  is  required  to  separate  them.  This  cele- 
brated experiment  was  devised  by  Otto  von  Guerike,  the  inven- 
tor of  the  air  pump,  and  it  was  performed  publicly  in  Magdeburg 
in  1654. 

(c)  The  hydrostatic  press  consists  essentially  of  a  strong  cylin- 
der with  a  large  plunger  or  piston,  and  a  pump  with  a  small  pis- 
ton or  plunger  for  forcing  water  into  the  large  cylinder  under 
high  pressure.  The  great  forging  press  at  the  Bethlehem  Steel 
Works  has  two  plungers  each  fifty  inches  in  diameter,  thus  ex- 
posing a  total  of  about  3,600  square  inches  of  piston  area  to  the 
water,  which  is  forced  into  the  cylinders  of  the  press  at  a  pres- 
sure of  8,000  pounds  per  square  inch.  This  gives  a  total  force 
of  about  14,000  tons  upon  the  two  plungers. 

110.  The  circumferential  tension  in  the  walls  of  a  cylindrical 
pipe.  —  The  pressure  of  a  fluid  in  a  cylindrical  pipe  produces  a 


Fig.   107. 


Fig.  108. 


Fig.  109. 


tension  in  the  material  of  the  pipe.  Consider  a  narrow  band  b, 
[Fig.  107,  of  the  material  of  a  pipe,  the  width  of  the  band  being 

me  inch.  An  end  view  of  this  band  is  shown  in  Fig.  108,  and, 
[since  the  band  is  one  inch  wide,  each  inch  of  its  circumference  is 
[pushed  outwards  by  a  force  equal  to  /  pounds,  where  p  is  the 
[steam  or  water  pressure  in  pounds  per  square  inch.     Therefore, 

iccording  to  Art.  60,  the  circumferential  tension  in  the  band  b  is 
[equal  to  rp  pounds  per  inch  of  width,  where  r  is  the  radius  of 
[the  pipe  in  inches. 


200.  ELEMENTS   OF  MECHANICS. 

It  is  instructive  to  establish  this  result  from  another  point  of 
view  as  follows  :  Imagine  the  cylindrical  pipe  to  be  half  solid,  as 
shown  by  the  shaded  area  in  Fig.  109,  then,  considering  one  inch 
of  length  of  the  pipe  as  before,  the  area  of  the  flat  surface  ab  is 
2r,  the  force  acting  on  this  flat  face  is  2rp,  and  this  force  is 
balanced  by  the  two  forces  FF^  so  that  the  value  of  each  force 
i^is  equal  to    rp. 

Since  the  circumferential  tension  in  a  cylindrical  pipe  is  equal 
to  r/,  it  is  evident  that  a  small  pipe  can  withstand  a  much 
greater  pressure  than  a  large  pipe,  the  thickness  of  the  walls  of 
the  pipe  being  the  same. 

111.  Pressure  in  a  liquid  due  to  gravity.  —  The  pressure  in  a 
fluid  under  the  action  of  gravity  increases  with  the  depth.  If  the 
density  of  the  fluid  is  the  same  throughout,  and  this  is  approxi- 
mately the  case  in  any  liquid,  then  the  pressure  at  a  point  distant 
X  centimeters  beneath  the  surface  of  the  liquid  exceeds  the  pres- 
sure at  the  surface  by  the  amount 

p  =  xdg  (60) 

in  which  /  is  expressed  in  dynes  per  square  centimeter,  d  is  the 
density  of  the  liquid  in  grams  per  cubic  centimeter,  and  g  is  the 
acceleration  of  gravity  in  centimeters  per  second  per  second.  If 
the  factor  g  is  omitted,  this  equation  gives  the  value  of  /  in 
grams-weight  per  square  centimeter. 

The  pressure  at  a  point  x  feet  beneath  the  surface  of  water  ex- 
ceeds the  pressure  at  the  surface  by  the  amount/  =  0.434;^,  where 
p  is  expressed  in  pounds-weight  per  square  inch.  The  factor 
0.434  is  the  weight  in  pounds  of  a  prism  of  water  one  foot  long 
and  one  square  inch  base,  that  is,  this  factor  is  the  density  of 
water  in  pounds  per  inch^-foot,  and  when  multiplied  by  feet  it 
gives  pounds  per  square  inch. 

Discussion  of  equation  (60). — The  force  with  which  a  liquid 
acts  on  an  element  of  an  exposed  surface  is  independent  of  the 
direction  of  the  element.*     Therefore  we  may  derive  equation 

*  See  the  discussion  of  Pascal's  principle  in  Art.  97. 


HYDROSTATICS. 


20I 


(60)  by  considering  a  horizontal  surface  of  area  a  exposed  to 
the  action  of  the  Hquid  as  shown  in  Fig.  1 10.  The  volume 
of  liquid  directly  above  a  is  ax^  the  mass  of  this  portion  of 
liquid  is  axd^  the  force  in  dynes  with  which  gravity  pulls  on  this 
portion  of  liquid  is  axdg^  and  therefore  the  total  force  with  which 
this  portion  of  liquid  pushes  down  on  the  element  a  is  equal  to 
axdg  dynes,  so  that  the  force  per  unit  area  is  axdg  divided  by  a. 
Equation  (60)  involves  no  consideration  of  the  shape  of  the 
vessel  which  contains  the  liquid.  As  a  matter  of  fact  the  pres- 
sure at  a  point  in  a  liquid  exceeds  the  pressure  at  the  surface  of 
the  liquid  by  the  amount   xdg   whatever  the  shape  and  size  of 


BBB 

pTT— f| 

ir^-Xv^^^>t5^i["fe^j^3^ 

i ""-""_-  "i  "  r"^"!^-"  J"  1  \_  ~ .  ""_"  : 

w 

&, 

}^'?-^A 

-■f-r'-f-^:  -<^lJ7^^ZK 

Fig.  110. 


Fig.  Ill, 


the  containing  vessel  may  be.  This  may  be  made  almost  self- 
evident  as  follows  :  Given  a  point  p,  Fig.  1 1 1 ,  at  a  distance  x 
beneath  the  surface  of  a  large  body  of  liquid.  Imagine  a  por- 
tion of  the  liquid  AAA  A,  of  any  shape  whatever,  extending  from 
p  to  the  surface.  The  liquid  surrounding  the  portion  A  AAA 
acts  on  AAA  A  exactly  as  a  containing  vessel  of  the  same  shape 
would  act,  and  therefore  the  pressure  of  /  is  exactly  what  it 
would  be  if  the  portion  AAA  A  were  contained  in  such  a  vessel. 
112.  The  total  force  acting  on  a  water  gate  and  its  point  of  ap- 
plication. —  When  a  plane  surface  of  area  a  is  exposed  to  the 
action  of  a  fluid  under  uniform  pressure,  the  total  force  acting  on 
the  surface  is  pa  and  the  point  of  application  of  this  force  is  the 
center  of  figure  of  the  exposed  plane  surface.     When,  however. 


202 


ELEMENTS   OF  MECHANICS. 


a  plane  surface  is  exposed  to  the  action  of  a  fluid  in  which  the 
pressure  is  not  uniform,  the  total  force  is,  of  course,  not  equal  to 
pa,    for  /  has  different  values  at  different  parts  of  the  surface, 


Fig.  112*. 


Fig.  1123. 


and  the  point  of  application  of  the  total  force  is  not  at  the  center 
of  figure  of  the  exposed  surface.  The  simplest  case  is  that  in 
which  the  water  in  a  tank  pushes  against  the  rectangular  side  of 

the  tank,  or  the  case  in 
which  water  pushes  against 
a  rectangular  gate  as  shown 
in,  Fig.  1 1 2a.  The  pressure 
at  the  top  of  the  gate  is 
/'  =  0.434.^'  pounds  per 
square  inch,  the  pressure  at 
the  bottom  of  the  gate  is 
/"  =  0.434;^",  the  average 
pressure  over  the  whole 
gate  is  (/'  +  /O/2  or 
Fig.  113.  0.434(;t:'  +  ;r'0/2    pounds 

per  square  inch,  and  the 
total  force  F  acting  on  the  gate  is  equal  to  the  product  of  this 
average  pressure  and  the  area  of  the  gate  in  square  inches. 


HYDROSTATICS.  203 

The  point  of  application  of  the  total  force  with  which  the  water 
pushes  on  the  gate  is  the  point  at  which  a  single  force  F\ 
Fig.  113,  could  be  applied  to  balance  the  push  of  the  water. 
This  point  is  evidently  below  the  center  of  the  gate ;  in  fact  the 
distance  X,  Fig.  113,  is 

In  the  case  of  the  side  of  a  rectangular  tank,  or  in  case  of  a 
dam  (where  x'  equals  zero)  the  distance  from  the  surface  of  the 
water  to  the  point  of  application  of  the  total  force  which  pushes 
on  the  side  of  the  tank  or  against  the  dam  is  two  thirds  of  the 
depth  of  the  water. 

Proof  of  equation  {61). — The  total  force  P,  Fig.  1 13,  is  equal  to  the  area  of  the 
gate  w{x^^  —  x^)  multiplied  by  the  average  pressure  o.424{x^^  -j-  x^)  /2,  w  being 
the  horizontal  width  of  the  gate  ;  and  the  torque  action  of  P  about  any  conveniently 
chosen  point  is  equal  to  the  sum  of  the  torque  actions,  about  the  same  point,  of  the 
forces  acting  on  the  various  elements  of  the  surface  of  the  gate.  Consider  a  horizontal 
strip  of  the  gate  distant  x  beneath  the  surface  of  the  water  and  of  which  the  vertical 
breadth  is  dx.  The  force  acting  on  this  strip  is  0.434^  X  w^jit,  and  the  torque  action 
of  this  force  about  a  point  at  the  surface  of  the  water  (lever  arm  x)  is  0.434x2  X^^/-^. 
Therefore  the  total  torque  action,  about  the  chosen  point,  of  the  forces  acting  on  the 
gate  is  equal  to 

o.4S47vj^^  .arVjir=o.434w  X  U^^^^  —  ^^^) 

whence,  placing  this  equal  to  the  torque  action    XF \^=Xy<(^0. 4^410 {x^^^  —  •^''^)Xi]> 
we  have  equation  (61). 

113.  Measurement  of  pressure.  The  barometer.  —  The  barom- 
eter consists  of  a  glass  tube  T,  Fig.  114,  filled  with  mercury  and 
inverted  in  an  open  vessel  of  mercury  CC,  the  tube  being  of 
such  length  that  an  empty  space  V  is  left  in  which  the  pressure 
is  zero.*  The  pressure  in  the  tube  at  the  level  of  the  mercury 
in  the  open  vessel  is  equal  to  atmospheric  pressure,  and  it  exceeds 
the  pressure  in  the  region  V  by  the  amount  xd^  according  to 
equation  (60).     Therefore,  since  the  pressure  in   V  is  zero,  the 

*  Even  if  the  tube  is  filled  with  extreme  care  so  as  to  exclude  all  of  the  air,  mer- 
cury vapor  will  form  in  the  region  F'and  the  pressure  will  not  be  exactly  zero. 


204 


ELEMENTS   OF   MECHANICS. 


air.; 


value  of  atmospheric  pressure  is  equal  to  xdg.  This  expression 
gives  the  value  of  atmospheric  pressure  in  dynes  per  square  cen- 
timeter, X  being  in  centimeters,  d  being  the  density  of  the  mer- 
cury in  grams  per  cubic  centimeter,  and  g  being  the  acceleration 
of  gravity  in  centimeters  per  second  per  second. 

If  the  mercury  is  at  some  standard 
temperature,  d  is  invariable  ;  and  if  the 
barometer  is  used  in  a  given  locality, 
g  is  invariable  ;  and  Jinder  these  condi- 
tions the  distance  x  may  be  used  as  a 
measure  of  the  pressure.     In  fact,  at- 
mospheric   pressure    is    usually    ex- 
pressed in    terms  of  the    height   the 
barometric    column    would    have    in 
millimeters  or  in  inches  if  the  mercury 
were  at  o°  C.  and  if  the  value  of  the 
acceleration   of  gravity    were    981.61 
cm./sec^  (its  value  at  45°  north  lati- 
tude at  sea  level).     To  facilitate  the 
accurate  use  of  the  barometer  in  dif- 
ferent localities  and  at  different  tem- 
peratures,   tables  *    have    been    pub- 
lished, with  the  help  of  which  the  height  of  barometric  column 
under  standard  conditions  as  to  temperature  and  gravity  may  be 
easily  found  from  its  observed  height  under  known  conditions. 

114.  Measurement  of  pressure.  Manometers  or  pressure  gauges. 
—  The  barometer  is  used  for  the  measurement  of  the  pressure 
of  the  atmosphere.  An  instrument  for  measuring  the  difference 
between  the  pressure  in  a  closed  vessel  and  atmospheric  pressure 
is  called  a  manometer  or  a  pressure  gauge. 

The  open  tube  manometer.  —  When  the  pressure  to  be  meas- 
ured is  small,  for  example,  when  it  is  desired  to  measure  the 


Fig.  1 14. 


*To  be  found  in  many  laboratory  reference  books.  For  example,  in  Kohlrausch's 
Physical  Measurements^  and  in  Landolt  and  Bornstein's  Physikalisch-Chemische 
Tabellen. 


HYDROSTATICS. 


205 


pressure  of  the  gases  at  the  base  of  a  smoke-stack,  or  the  pres- 
sure developed  by  a  fan  blower,  the  pressure  is  determined  by 
measuring  the  height  of  water  or  mercury  column  which  it  will 
support.     Thus  Fig.  1 1 5 

shows     an     open    tube  ,••'.:" /'I  v-l^ '*:•>:"'£ 

manometer  arranged  fDr  *^,'^<'^::'•W;^:: ''■/":' 

measuring  the    pressure  '^S- 

of  the  gas  in  city  mains.  "•"*/*.  v., 'air;._.;.  ••  afr. 

The  Bourdon  gauge. 
—  The  pressure  gauge 
commonly  used  on  steam  p '  ■  . .  y~ 
boilers  is  usually  of  the 
type  known  as  the  Bour- 
don gauge,  of  which  the 
essential  features  are 
shown  in  Fig.  116.  A 
very  thin  walled  metal 
tube  abc  of  flat  elliptical 
section  is  closed  at  the  end  ^,  and  the  end  a  communicates 
through  the  tube  tt  with  the  steam  boiler.     The  pressure  inside 

h 


T 


Fig.  115. 


to  gSLVL&S 


Fig.  116. 


Fig.  117. 


of  the  tube  abc  tends  to  straighten  it,  and  the  movement  of  the 
end  c  actuates  a  pointer  which  plays  over  a  scale  the  divisions 


206  '  ELEMENTS   OF   MECHANICS. 

of  which  are  determined  by  calibration,  that  is,  by  noting  the 
position  of  the  pointer  for  various  known  pressures. 

The  gauge  tester  is  a  device  for  generating  accurately  known 
pressures  which  are  communicated  to  a  pressure  gauge  which  is 
to  be  calibrated.  It  consists  of  a  small  metal  chamber  filled  with 
oil.  A  plunger  of  known  area  a  is  forced  into  this  chamber  by 
a  known  weight,  and  the  known  pressure  thus  developed  is  com- 
municated to  the  gauge. 

115.  Buoyant  force  of  fluids.  — A  body  which  is  submerged  in 
a  fluid  is  pushed  upwards  by  a  force  which  is  equal  to  the  weight 
of  its  volume  of  the  fluid.  The  point  of  application  of  this  force 
is  the  center  of  figure  *  of  the  submerged  body  and  it  is  called 
the  center  of  buoyancy.  This  principle  was  first  enunciated  by 
Archimedes,  and  it  is  called  Archimedes'  principle.  It  may  be 
made  almost  self-evident  by  the  following  considerations  :  Given 
a  fluid  at  rest.  Imagine  a  certain  portion  of  this  fluid  of  any 
size  and  shape.  This  portion  is  stationary,  and  therefore  the  sur- 
rounding fluid  pushes  upwards  upon  it  with  a  force  which  is  equal 
to  its  weight,  and  the  point  of  application  of  this  upward  force 
is  the  center  of  mass  of  the  portion.  But  the  surrounding  fluid 
acts  upon  the  given  portion  in  exactly  the  same  way  that  it 
would  act  upon  a  submerged  body  of  the  same  size  and  shape. 

A  body  which  is  partly  submerged  in  a  liquid  is  pushed  up- 
wards by  a  force  which  is  equal  to  the  weight  of  the  displaced 
volume  of  liquid,  and  the  point  of  the  application  of  this  upward 
force  is  the  center  of  figure  of  the  submerged  part  of  the  body. 
Therefore  a  floating  body  displaces  its  weight  of  the  liquid  in 
which  it  floats. 

Examples.  —  The  buoyant  force  of  water  is  familiar  in  a  general 
way  to  everyone.  The  principle  of  Archimedes  is  utilized  in  the 
ordinary  method  of  finding  the  specific  gravity  of  a  body  as  fol- 
lows :  The  body  is  weighed  in  the  air  and  then  it  is  suspended 
under  water  and  weighed  again.  The  difference  is  the  weight 
(mass)  of  its  volume  of  water,  and  the  specific  gravity  of  the  body 

*  The  center  of  mass  of  the  body,  if  the  body  is  homogeneous. 


HYDROSTATICS.  20/ 

may  then  be  calculated  according  to  the  principles  enunciated  in 
Art.  8. 

When  a  body  is  weighed  on  a  balance  its  weight  (mass)  is  un- 
derestimated if  it  is  more  bulky  than  the  weights  that  are  used 
to  balance  it ;  this  is  on  account  of  the  greater  buoyant  force  ex- 
erted by  the  air  on  the  body  than  on  the  weights.  This  error  is 
often  quite  appreciable,  and  it  must  be  allowed  for  in  accurate 
weighing. 

The  buoyant  force  of  the  air  is  most  strikingly  shown  by  the 
balloon. 

116.  Equilibrium  of  a  floating  body.*  —  A  body  is  said  to  be 
in  unstable  equilibrium  when  the  forces  which  act  upon  it  tend  to 
carry  it  farther  and  farther  from  its  equilibrium  position  when  it  is 
displaced  slightly  therefrom.  Thus  a  body  standing  vertically  on 
a  sharp  point  is  in  unstable  equilibrium,  the  least  displacement  of 
the  body  in  any  direction  causes  it  to  fall  over. 

A  body  is  said  to  be  in  neutral  equilibrium  when  the  forces 
which  act  upon  the  body  remain  in  equilibrium  as  the  body 
moves.  Thus  a  homogeneous  sphere  resting  on  a  smooth  hori- 
zontal table,  and  a  balanced  wheel  supported  on  an  axle  are  in 
neutral  equilibrium. 

A  body  is  said  to  be  in  stable  equilibrium,  when  the  forces  which 
act  upon  it  tend  to  bring  it  back  to  its  equilibrium  position  when 
it  is  displaced  therefrom.  Thus  a  weight  fixed  to  the  end  of  a 
spring,  a  pendulum  hanging  vertically  downwards,  and  a  block 
resting  on  a  table  are  in  stable  equilibrium. 

A  body  is  said  to  have  a  high  degree  of  stability  when  a  very 
considerable  force  is  required  to  displace  it  from  its  equilibrium 
position.  Thus  a  broad  sail-boat  with  its  ballast  placed  low  down 
in  its  hold  is  very  stable,  because  a  very  considerable  force  is  re- 
quired to  turn  the  boat  from  its  vertical  position. 

Condition  of  equilibrium  of  a  floating  body.  —  When  a  floating 
body  is  stationary,  it  is,  of  course,  in  equilibrium  and  the  down- 
ward force  of  gravity  must  have  the  same  line  of  action  as  the 

*This  subject  is  treated  in  detail  in  works  on  naval  architecture. 


2o8 


ELEMENTS   OF   MECHANICS. 


upward  force  of  buoyancy,  otherwise  these  two  forces  would  have 
an  unbalanced  torque  action  and  the  body  would  not  be  in  equi- 
librium. Therefore  the  center  of  a  mass  of  a  floating  body  and 
the  center  of  figure  of  the  submerged  portion  of  the  body  {center 
of  buoyancy^  m,ust  lie  in  the  same  vertical  line. 

The  problem  of  determining  the  degree  of  stability  of  a  floating 
body  is  greatly  complicated  by  the  change  of  shape  of  the  sub- 
merged part  of  the  body  when  the  body  is  tilted  to  one  side, 
and  the  shifting  of  the  center  of  buoyancy  which  is  due  to  this 


Fig.  119. 


change  of  shape.  Therefore  the  simplest  case  is  that  of  a  floating 
body  of  which  the  submerged  portion  does  not  change  its  shape 
when  the  body  is  tilted  to  one  side. 

Examples  of  simplest  case. — The  submerged  part  of  a  floating 
sphere  is  the  same  in  shape  however  the  sphere  be  turned  and 
therefore  the  center  of  buoyancy  does  not  move  as  the  sphere  is 
turned.     If  the  center  of  mass  of  the  sphere  is  at  its  geometrical 


HYDROSTATICS. 


209 


center  we  have  a  case  of  neutral  equilibrium  of  floating ;  if  the 
sphere  is  heavier  on  one  side,  it  floats  in  stable  equilibrium  with 
its  heavy  side  downwards,  and  in  unstable  equihbrium  with  its 
heavy  side  upwards. 

The  most  interesting  simple  example  of  equilibrium  of  floating 


Fig.  120. 

is  the  hydrometer  as  shown  in  Figs.  1 1 8  and  1 1 9.  The  shape 
of  the  submerged  portion  is  slightly  altered  when  the  hydrometer 
is  tipped  over,  but  the  change  of  shape  is  nearly  negligible  and 
the  center  of  buoyancy  b  is  nearly  fixed  in  position. 

Example  of  the  general  case,  —  Consider  two  floats  A  and  B^ 


Figs.  1 20  and  121,  connected  rigidly  together  by  a  beam.  This 
arrangement  is  similar  to  the  style  of  boat  called  a  catamaran, 
and  when  it  is  in  equilibrium  the  center  of  mass  m  and  the  center 
of  buoyancy  b  are  located  as  show^n  in  Fig.  1 20.  When,  how- 
ever, the  arrangement  is  tilted,  as  shown  in  Fig.  121,  the  center 
of  buoyancy  b  shifts  towards  the  lower  side,  while  the  center  of 
14 


2IO 


ELEMENTS   OF   MECHANICS. 


mass  fn  of  course  remains  stationary.  The  arrangement  behaves^ 
for  slight  angles  of  tilting^  as  if  its  center  of  buoyancy  were  fixed 
in  the  line  ma  at  the  place  B  where  the  line  m,a  is  cut  by  the 
vertical  line  be  in  Fig.  121,  because  the  line  of  action  of  the 
buoyant  force  passes  through  the  point  B  for  any  small  angle  of 
tilting.     The  point  B  is  called  the  metacenter  of  the  float. ' 

117.  The  hydrometer. — The  common  form  of  the  hydrometer 
is  a  light  glass  float,  weighted  at  one  end  with  lead  or  mercury, 

and  having  a  cylindrical  glass  stem  at 
the  other  end  as  shown  in  Fig.  118. 
This  float  sinks  to  different  depths  in 
liquids  of  different  specific  gravities  and 
upon  the  stem  is  a  scale  which  indicates 
the  specific  gravity  of  the  liquid  in  which 
the  instrument  is  placed. 

The  specific  gravity  scale.  —  To  con- 
struct a  specific  gravity  scale  on  the 
stem  of  a  hydrometer,  the  instrument 
is  floated  in  water  and  the  water-mark 
located,  the  instrument  is  then  floated 
in  a  liquid  of  known  specific  gravity  «, 
the  <3:-mark  is  located,  and  then  the  dis- 
tance /  between  the  water-mark  and 
the  ^-mark  is  measured.  The  scale  is 
then  determined  by  calculating  the  dis- 
tance from  the  water-mark  to  each  desired  mark  of  the  scale. 
Thus  the  distance  d  from  the  water-mark  to  the  ^-mark  {s  being 
a  specific  gravity,  i.io,  1.20,  etc.)  is  given  by  the  formula 


/-N 

-wd-Ut-msLtk 

! 

d 

j 

s-nmrk 

i 

a-mark 



k, 

T 


J._. 


Fig.  122. 


I  — 


d=l 


(62) 


This  equation  may  be  derived  as  follows :  A  floating  body  dis- 
places its  weight  of  a  liquid.     The  volume  of  water  displaced  by 


HYDROSTATICS.  211 

the  instrument  being  taken  as  unity,  the  volume  below  the  ^-rnark 
is  I  la  and  the  volume  below  the  j-mark  is  i  js  inasmuch  as  these 
liquids  are  supposed  to  be  a  times  as  heavy  and  s  times  as  heavy 
as  water  respectively.  Therefore  the  volume  of  the  length  /  of 
the  stem  is  (i  —  i  ja)  and  the  volume  of  the  length  d  of  the  stem 
is  (i  —  1/5),  and,  the  stem  being  assumed  cylindrical,  the  lengths 
/  and  d  are  proportional  to  these  volumes. 

Beaume  hydrometer  scales.  —  The  specific  gravity  scale  on  a 
hydrometer  is  not  a  scale  of  equal  parts  and  therefore  the  con- 
struction of  the  scale  is  tedious.  One  account  of  this  fact  a  num- 
ber of  schemes  have  been  proposed  for  constructing  hydrometers 
with  arbitrary  scales  of  equal  parts.  Of  these  scales  those  of 
Beaume  are  most  extensively  used. 

Beaume' s  scale  for  heavy  liquids  is  constructed  by  locating  the 
water-mark  (near  the  top  of  the  stem),  and  the  mark  to  which 
the  instrument  sinks  in  a  1 5  per  cent,  solution  of  pure  sodium 
chloride  (common  salt).  The  space  between  these  marks  is 
divided  into  15  equal  parts,  and  divisions  of  like  size  are  con- 
tinued down  the  stem.  These  divisions  are  numbered  down- 
wards from  the  water-mark.  A  liquid  is  said  to  have  a  specific 
gravity  of  26°  Beaume  heavy  when  the  hydrometer  sinks  in  it  to 
mark  number  twenty-six  on  the  scale  here  described. 

Beaume's  scale  for  light  liquids  is  constructed  by  locating  the 
mark  to  which  the  instrument  sinks  in  a  10  per  cent,  solution  of 
sodium  chloride  (near  the  bottom  of  the  stem),  and  the  water- 
mark. The  space  between  ^hese  marks  is  divided  into  10  equal 
parts,  and  divisions  of  like  size  are  continued  up  the  stem.  These 
divisions  are  numbered  upwards  from  the  salt  solution  mark.  A 
liquid  is  said  to  have  a  specific  gravity  of  1 7  °  Beaume  light  when 
the  hydrometer  sinks  to  mark  number  seventeen  on  the  scale  here 
described. 

CAPILLARY   PHENOMENA   OF   LIQUIDS. 
118.  Cohesion;  adhesion.  —  When  a  body  is  under  stress,  as 
for  example  a  stretched  wire,  the  tendency  of  the  stress  is  to  tear 


212     •  ELEMENTS    OF   MECHANICS. 

the  contiguous  parts  of  the  body  asunder.  The  forces  which  op- 
pose this  tendency  and  hold  the  contiguous  parts  of  a  body  to- 
gether are  called  the  forces  of  cohesion.  The  forces  which  cause 
dissimilar  substances  to  cling  together  are  called  the  forces  of 
adhesion.  The  discussion  of  the  elastic  properties  of  solids  is  a 
discussion  of  their  properties  of  cohesion.  The  cohesion  of  water 
and  the  adhesion  between  water  and  glass  are  the  forces  which 
determine  the  curious  behavior  of  water  in  a  fine  hair-like  tube 
of  glass,  and  the  phenomena  exhibited  by  liquids  because  of  co- 
hesion and  adhesion  are  called  capillary  phenomena  from  the 
Latin  word  capillaris  meaning  a  hair. 

119.  Surface  tension.  —  On  account  of  their  cohesion,  all 
liquids  behave  as  if  their  free  surfaces  were  stretched  skins,  that 
is,  as  if  their  free  surfaces  were  under  tension.  Thus  a  drop  of  a 
liquid  tends  to  assume  a  spherical  shape  on  account  of  its  surface 
tension.  A  mixture  of  water  and  alcohol  may  be  made  of  the 
same  density  as  olive  oil,  and  a  drop  of  olive  oil  suspended  in 
such  a  mixture  becomes  perfectly  spherical. 

Many  curious  phenomena  *  are  produced  by  the  variation  of 
the  surface  tension  of  a  liquid  with  admixture  of  other  hquids  or 
with  temperature.  Thus  a  drop  of  kerosene  spreads  out  in  an 
ever  widening  layer  on  a  clean  water  surface,  on  account  of  the 
fact  that  the  tension  of  the  clean  water  surface  beyond  the  layer 
of  oil  is  greater  than  the  tension  of  the  oily  surface.  A  small 
shaving  of  camphor  gum  darts  about  in  a  very  striking  way  upon 
a  clean  water  surface,  on  account  of  the  fact  that  the  camphor 
dissolves  in  the  water  more  rapidly  where  the  shaving  happens 
to  have  a  sharp  projecting  point,  the  water  surface  has  a  lessened 
tension  where  the  camphor  dissolves,  and  the  greater  tension  on 
the  opposite  side  pulls  the  shaving  along.  A  thin  layer  of  water 
on  a  horizontal  glass  plate  draws  itself  away  and  leaves  a  dry 
spot  where  a  drop  of  alcohol  is  let  fall  on  the  plate.  A  thin 
layer  of  lard  on  the  bottom  of  a  frying  pan  pulls  itself  away  from 

*  See  the  very  interesting  article  capillary  action  in  the  Encyclopedia  Britannica. 
This  article  also  gives  a  comprehensive  discussion  of  the  theory  of  capillary  action. 


HYDROSTATICS. 


213 


the  hotter  parts  of  the  pan  and  heaps  itself  up  on  the  cooler 
parts,  because  of  the  greater  surface  tension  of  the  cooler  lard. 

120.  Angles  of  contact.  Capillary  elevation  and  depression.  — 
The  clean  surface  of  a  liquid  always  meets  the  clean  walls  of  a 
containing  vessel  at  a  definite  angle.  Thus  a  clean  surface  of 
water  turns  upwards  and  meets  a  clean  glass  wall  tangentially, 
and  a  clean  surface  of  mercury  turns  downwards  and  meets  a 
clean  glass  wall  at  an  angle  of  51°  8'. 

Since  a  clean  water  surface  turns  upwards  and  meets  a  glass 
wall  tangentially  it  is  evident  that  the  surface  of  water  in  a  small 
glass  tube  must  be  concave  as  shown  in  Fig.  123,  and  the  result 


Fig.  123. 


Fig.  124. 


is  that  the  water  is  drawn  up  into  the  tube.  On  the  other  hand, 
the  surface  of  mercury  in  a  small  glass  tube  is  convex  and  the 
surface  tension  pulls  the  mercury  down  below  the  level  of  the 
surrounding  mercury  as  shown  in  Fig.  124. 

121.  Measurement  of  surface  tension  of  water.  —  Let  r  be  the 
radius  of  the  bore  of  the  glass  tube  in  Fig.  123.  Then  the  cir- 
cumference 27rr  is  the  width  of  the  surface  film  of  water  at  the 
point  of  tangency,  and  2'TrrT  is  the  total  upward  force  due  to 
the  tension  of  the  film,  T  being  the  tension  per  unit  width.  The 
volume  of  water  in  the  tube  above  the  level  of  the  surrounding 
water  is  irr'^h,  and  the  weight  of  this  water  is  irr^hdg  where  d 
is  the  density  in  grams  per  cubic  centimeter  and  g  is  the  accelera- 


214 


ELEMENTS   OF   MECHANICS. 


tion  of  gravity.      The  weight  of  water  in  the  tube  being  sup- 
ported by  the  tension  of  the  film,  we  have 


whence 


27rr7"=  irr^hdg 


J      rhdg 


from  which  T  may  be  calculated  when  r,  </,  and  g  are  known 
and  h  observed.  The  surface  tension  of  water  is  found  in  this 
way  to  be  8 1  dynes  per  centimeter  breadth. 

Problems. 

136.  Calculate  the  number  of  dynes  per  square  centimeter  in 
one  pound-weight  per  square  inch,  taking  the  acceleration  of 
gravity  equal  to  980  cm./sec^. 

137.  Figure   137/  represents  a  hydrostatic  press.     The   dis- 


iffater 


"^ 


iiiiiimiiiiiiJ 


Fig.  137A 

tances  a  and  b  are  equal  to  6  inches  and  6  feet  respectively,  the 
diameter  of  the  pump  plunger  /is  1.5  inches  and  the  diameter 
of  the  press  plunger  P  is  24  inches.  Find  the  total  force  on  P 
due  to  a  force  of  100  pounds  at  F,  neglecting  friction. 

138.  Calculate  the  circumferential  tension  in  the  cylindrical 
shell  of  a  boiler  due  to  a  steam  pressure  of  125  pounds  per 
square  inch,  the  diameter  of  the  boiler  being  6  feet. 

139.  Sheet  steel  0.02  inch  thick  will  safely  stand  a  tension  of 
200  pounds  per  inch  of  width.     What  is  the  greatest  diameter 


HYDROSTATICS. 


215 


of  steel  tube  with  0.02  inch  wall,  which  can  safely  withstand  a 
pressure  of  1 50  pounds  per  square  inch  ? 

140.  Calculate  the  pressure  of  the  air  in  the  caisson  shown  in 
Fig.  140/ ;  the  distance  from  the  water  level  in  the  river  to  the 
water  level  in  the  caisson  being  90  feet. 

141.  Oil  and  water  are  drawn  up  in  two  connecting  tubes  as 
shown  in  Fig.  141/.     The  height  of  the  water  column  is   36 


.#fr 


oil  awater 


Fig.  141/. 


Fig.  142A 


inches,  and  the  height  of  the  oil  column  is  42  inches.     What  is  the 
ratio  of  the  densities  of  oil  and  water  (specific  gravity  of  the  oil)  ? 

142.  The  density  of  air  under  ordinary  conditions  is  0.0012 
grams  per  cubic  centimeter,  and  the  density  of  illuminating  gas 
is,  say,  0.0008  grams  per  cubic  centimeter.  The  pressure  of 
illuminating  gas  at  the  base  of  the  gas  holder  exceeds  the  pres- 
sure of  the  outside  air  at  the  same  level  by  2  inches  of  water. 
Find  the  difference  between  the  gas  and  air  pressures  on  top  of  a 
hill  at  a  height  /i  of  400  feet  above  the  gas  holder. 

ATote.  —  In  this  problem  assume  that  the  density  of  the  gas  is  uniform  and  that 
the  density  of  the  air  is  uniform.  The  density  of  water  is  one  gram  per  cubic  centi- 
meter and  one  inch  equals  2.54  centimeters. 

143.  Sketch  the  form  of  a  dish  such  that  the  total  force  due  to 
hydrostatic  pressure  on  its  bottom  shall  be  (a)  greater  than,  (^) 
equal  to,  and  (c)  less  than  the  weight  of  the  contained  liquid. 

144^.  The  density  of  mercury  at  0°  C.  is  13.5956  grams  per 
cubic  centimeter.  Calculate  the  value  in  dynes  per  square  centi- 
meter of  standard  atmospheric  pressure,  namely  y6  cm.  of  mer- 


2l6  ELEMENTS    OF   MECHANICS. 

cury  at  o°  C,  the  value  of  gravity  being  980.61  cm.  per  second 
per  second.     Give  the  result  in  dynes  per  square  centimeter. 

144^.  The  specific  gravity  of  mercury  is  approximately  13.6. 
The  pressure  in  pounds  per  square  inch  at  a  point  x  feet  beneath 
pure  water  is  /  =  0.434X  Find  the  value  in  pounds  per  square 
inch  of  one  English  standard  atmosphere,  namely,  30  inches  of 
mercury. 

145.  Calculate  the  height  of  the  homogeneous  atmosphere ;  that 
is,  assuming  that  the  atmosphere  has  a  uniform  density  of 
0.00129  grams  per  cubic  centimeter  throughout,  calculate  the 
depth  which  would  produce  standard  atmospheric  pressure. 

146.  A  masonry  dam  1 2  feet  high  is  to  be  built  on  a  flat  rock 
bottom.  The  dam  is  to  have  the  shape  of  a  rectangular  paral- 
lelopiped  (in  order  that  this  problem  may  be  simple).  How  thick 
must  the  dam  be  made  so  that  the  force  of  the  water  will  just  be 
unable  to  tip  it  over?  The  masonry  weighs  120  pounds  per 
cubic  foot. 

Note.  —  The  moment  of  the  force  with  which  the  water  pushes  on  the  dam  is  to 
be  equal  to  the  moment  of  the  gravity- pull  on  the  dam,  both  moments  being  taken 
about  the  down-stream,  bottom  edge  of  the  dam. 

147.  A  water  gate  36  inches  wide  has  its  upper  edge  2  feet 
beneath  the  level  of  the  water  in  a  canal  lock,  and  its  lower  edge 
5  feet  beneath  the  water  level.  What  is  the  total  force  on  the 
gate  and  what  is  the  distance,  beneath  the  water  level,  of  the 
point  of  application  of  this  total  or  resultant  force  ? 

148.  A  piece  of  lead  weighs  233.60  grams  in  air  and  212.9 
grams  in  water  at  20°  C.  What  is  the  specific  gravity  and  the 
density  of  lead  at  20°  C.  ? 

For  table  of  densities  of  water  at  various  temperatures  see  chapters  on  heat, 

149.  A  piece  of  glass  weighs  260.7  grams  in  air  and  153.8 
grams  in  water  at  20°  C.  The  same  piece  of  glass  weighs  92.2 
grams  in  H2S0^  at  20°  C.  What  is  the  specific  gravity  of  HgSO^ 
at  20°  C.  ? 

150.  A  glass  bulb  weighs  75.405  grams  when  filled  with  air  at 
standard  temperature  and  pressure.     It  weighs  74.309   grams 


HYDROSTATICS.  21/ 

when  the  air  is  pumped  out.  It  weighs  74.422  grams  when 
filled  with  hydrogen  at  the  same  temperature  and  pressure. 
What  is  the  specific  gravity  of  hydrogen  referred  to  air  ? 

151.  What  is  the  net  lifting  capacity  of  a  balloon  containing 
400  cubic  meters  of  hydrogen,  its  material  weighing  250  kilo- 
grams ?  (Weight  of  a  cubic  meter  of  air  is  1,200  grams  ;  weight 
of  a  cubic  meter  of  hydrogen  is  90  grams.) 

152.  The  distance  along  a  hydrometer  stem  from  the  water 
mark  to  the  mark  to  which  the  instrument  sinks  in  kerosene 
(specific  gravity  0.79)  is  9.62  centimeters.  Calculate  the  distance 
from  the  water  mark  to  the  marks  to  which  the  instrument  would 
sink  in  a  20^  solution  of  alcohol,  in  a  40^  solution  of  alcohol,  in 
a  60^  solution  of  alcohol,  in  an  80^  solution  of  alcohol,  and  in 
pure  alcohol.  The  specific  gravities  of  these  solutions  are  as  fol- 
lows: 20/0  =  0.975;  40^  =  0.951;  60/0  =  0.913;  80/)  =  0.863; 
100/)  =  0.794. 

153.  The  specific  gravity  of  a  1 5  per  cent,  solution  of  sodium 
chloride  at  ordinary  room  temperature  is  1.1115.  Calculate  the 
specific  gravity  corresponding  to  26°  Beaume  (heavy). 

Note.  — This  problem  is  to  be  solved  by  using  equation  (62)  in  a  way  that  will 
be  apparent  when  it  is  considered  that  degrees  Beaum6  represent  distances  along  the 
hydrometer  stem. 

154.  The  specific  gravity  of  a  10  per  cent,  solution  of  sodium 
chloride  at  ordinary  room  temperature  is  1.0734.  Calculate  the 
specific  gravity  corresponding  to  20°  Beaume  (light). 

155.  Two  rectangular  boxes  12  inches  x  12  inches  x  16  feet 
are  fastened  to  two  cross-beams  like  a  catamaran,  and  the  whole 
weighs  625  pounds.  The  distance  apart  from  center  to  center  of 
the  two  boxes  is  6  feet.  Find  how  far  to  one  side  the  center  of 
buoyancy  shifts  when  the  raft  is  tilted  5°  about  its  longitudinal 
axis,  find  the  approximate  position  of  the  metacenter,  and  find 
the  torque  tending  to  bring  the  raft  into  a  horizontal  position. 

Note.  —  For  the  sake  of  simplicity  assume  that  the  submerged  part  of  each  box  re- 
mains rectangular.  The  metacenter  is  defined  in  terms  of  an  infinitesimal  angle  of 
tilting,  but  its  position  for  a  5°  tilt  may  be  determined  without  the  use  of  calculus  in 
the  case  here  considered. 


CHAPTER   X. 
HYDRAULICS. 

[Throughout  this  chapter  the  following  units  are  used :  feet, 
feet  per  second,  square  feet,  cubic  feet,  pounds  (mass),  pounds 
per  cubic  foot  (density),  pounds  (force),  pounds  per  square  foot 
(pressure),  and  foot-pounds  (energy).  The  factor  g  is  equal  to 
32.2  which  is  the  acceleration  in  feet  per  second  per  second  of  a 
one  pound  body  when  acted  upon  by  an  unbalanced  force  of  one 
pound- weight.] 

122.  Limitations  of  this  chapter.  —  Hydraulics,  in  the  general 
sense  in  which  the  term  is  here  used,  is  the  study  of  liquids  and 
gases  in  motion  ;  and  the  phenomena  which  are  presented  in  this 
branch  of  physics  are  excessively  complicated.  Even  the  appar- 
ently steady  flow  of  a  great  river  through  a  smooth  sandy  chan- 
nel is  an  endlessly  intricate  combination  of  boiling  and  whirling 
motion ;  and  the  jet  of  spray  from  a  hydrant,  or  the  burst  of 
steam  from  the  safety-valve  of  a  locomotive,  what  is  to  be  said  of 
such  things  as  these  ?  Or  let  one  consider  the  fitful  motion  of 
the  wind  as  indicated  by  the  swaying  of  trees  and  as  actually  vis- 
ible in  driven  clouds  of  dust  and  smoke,  or  the  sweep  of  the 
flames  in  a  conflagration  !  These  are  actual  examples  of  fluids 
in  motion,  and  they  are  indescribably,  infinitely  *  complicated. 
The  finer  details  of  such  phenomena,  however,  are  devoid   of 

*  Everyone  concedes  the  idea  of  infinity  which  is  based  upon  abstract  numerals 
(one,  two,  three,  four  and  so  on  ad  infinitum  !),  and  the  idea  of  infinity  which  is 
based  on  the  notion  of  a  straight  line  ;  but  most  men  are  wholly  concerned  with  the 
humanly  significant  and  persistent  phases  of  the  material  world,  their  perception  does 
penetrate  into  the  substratum  of  utterly  confused  and  erratic  action  which  underlies 
every  physical  phenomenon,  and  they  balk  at  the  suggestion  that  the  phenomena  of 
fluid  motion,  for  example,  are  infinitely  complicated.  Surely  the  abstract  idea  of  in- 
finity is  as  nothing  compared  with  the  awful  intimation  of  infinity  that  comes  from 
things  that  are  seen  and  felt. 

218 


HYDRAULICS.  219 

practical  significance,  indeed  they  present  but  little  that  is  suffici- 
ently definite  even  to  be  intelligible. 

The  science  of  hydraulics  is  based  on  ideas  which  refer  to  general 
aspects  of  fluid  motion^  like  a  sailor's  idea  of  a  ten-knot  wind  ;  and, 
indeed,  the  engineer  is  concerned  chiefly  with  what  may  be  called 
average  effects  such  as  the  time  required  to  draw  a  pail  of  water 
from  a  hydrant,  the  loss  of  pressure  in  a  line  of  pipe  between  a 
pump  and  a  fire  nozzle,  or  the  force  exerted  by  a  water  jet  on  the 
buckets  of  a  water  wheel.  These  are  called  average  effects  be- 
cause they  are  never  perfectly  steady  but  always  subject  to  per- 
ceptible fluctuations  of  an  erratic  character,  and  to  think  of  any 
of  these  effects  as  having  a  definite  value  is  of  course  to  think  of 
its  average  value  under  the  given  conditions.*  The  extent  to 
which  the  practical  science  of  hydraulics  is  limited  is  evident  from 
the  following  outline  of  the  ideal  types  of  flow  upon  which  nearly 
the  whole  of  the  science  is  based. 

Permanent  and  varyifig  states  of  flow.  —  When  a  hydrant  is 
suddenly  opened,  it  takes  an  appreciable  time  for  the  flow  of 
water  to  become  steady.  During  this  time  {a)  the  velocity  at  each 
point  of  the  stream  is  increasing  and  perhaps  changing  in  direc- 
tion also.  After  a  short  time,  however,  the  flow  becomes  fully 
established  and  then  (U)  the  velocity  at  each  point  in  the  stream  re- 
mains  unchanged  in  magnitude  and  direction.'\  The  motion  {a) 
is  called  a  varying  state  of  flow ^  and  the  motion  {U)  is  called  a 
permanent  state  of  flow.  Most  of  the  following  discussion  applies 
to  permanent  states  of  flow,  indeed  there  are  but  few  cases  in 
which  it  is  important  to  consider  varying  states  of  flow. 

The  idea  of  simple  flow.  Stream  lines.  —  The  idea  of  simple 
flow  applies  both  to  permanent  and  to  varying  states  of  flow,  but 
it  is  sufficient  to  explain  the  idea  in  its  application  to  permanent 
flow  only.     When  water  flows  steadily  through  a  pipe,  the  motion 

*  See  two  brief  articles  by  W.  S.  Franklin,  Transactions  of  American  Institute  of 
Electrical  Engineers,  Vol.  XX,  pages  285-286 ;  and  Science,  Vol.  XIV,  pages  496- 
497,  September  27,  1901. 

t  Assuming  the  stream  to  be  free  from  turbulence.  See  the  following  definition  of 
simple  flow. 


220 


ELEMENTS   OF   MECHANICS. 


is  always  more  or  less  complicated  by  continually  changing 
eddies,  the  water  at  a  given  point  does  not  continue  to  move  in  a 
fixed  direction  at  a  constant  velocity ;  nevertheless,  it  is  conveni- 
ent to  treat  the  motion  as  if  the  velocity  of  the  water  were  in  a 
fixed  direction  and  constant  in  magnitude  at  each  point.  Such  a 
motion  is  called  a  simple  flow.     In  the  case  of  a  simple  flow  a  line 


can  be  imagined  to  be  drawn  through  the  fluid  so  as  to  be  at 
each  point  in  the  direction  of  the  flow  at  that  point.  Such 
a  line  is  called  a  stream  line.  Thus  the  fine  lines  in  Fig.  125 
are  stream  lines  representing  a  simple  flow  of  water  through  a 
contracted  part  of  a  pipe.  To  apply  the  idea  of  simple  flow  to 
an  actual  case  of  fluid  motion  is  the  same  thing  as  to  consider 


Fig.  126. 

the  average  character  of  the  motion  during  a  fairly  long  interval 
of  time. 

Lamellar  flow.  —  Even  though  the  motion  of  water  in  a  pipe 
may  be  approximately  a  simple  flow,  the  velocity  may  not  be  the 
same  at  every  point  in  a  given  cross-section  of  the  pipe,  that  is, 
the  velocity  may  not  be  the  same  at  every  part  of  the  layer  aby 
Fig.  1 26 ;  in  fact  the  water  near  the  walls  always  moves  slower 
than  the  water  near  the  center  of  a  pipe  ;  nevertheless,  it  is  con- 
venient in  many  cases  to  treat  the  motion  as  if  the  velocity  were 
the  same  at  every  point  in  any  layer  like    ab,    Fig.  1 26.     Such  an 


HYDRAULICS. 


221 


ideal  flow  is  called  a  lamellar  Jlow,  because  in  such  a  flow  the 
fluid  in  any  layer  or  lamella  ab  would  later  be  found  in  the  layer 
cd,  and  still  later  in  the  layer  ef.  To  apply  the  idea  of  lamellar 
flow  to  an  actual  case  of  fluid  motion  is  the  same  thing  as  to  con- 
sider the  average  velocity  over  the  entire  cross-section  of  a  stream. 

Rotational  and  irrotational  Jlow.  — In  certain  cases  of  fluid  motion  each  particle  of 
the  fluid,  if  suddenly  solidified,  would  be  found  to  be  rotating  at  a  definite  angular 
velocity  about  a  definite  axis  ;  such  fluid  motion  is  called  rotational  tnotion  or  vortex 
motion.  Thus  the  whirling  motion  of  the  water  in  an  emptying  sink  is  vortex  motion. 
In  other  cases  of  fluid  motion  the  particles  of  the  fluid  are  not  rotating  ;  this  kind  of 
fluid  motion  is  called  irrotational  motion.  Some  of  the  important  practical  aspects 
of  vortex  motion  are  discussed  in  the  Encyclopedia  Britannica  article  Hydromechanics, 
Part  TIL,  Hydraulics,  sections  30,  31,  103  and  190. 

In  irrotational  fluid  motion  the  velocity  can  be  represented  as  a  potential  gradient, 
whereas  in  rotational  fluid  motion  the  velocity  cannot  be  represented  as  a  potential 
gradient.     See  Art.  21,  space  variation  of  vectors. 

123.  Some  actual  phenomena  of  fluid  motion.  —  The  following 
treatment  of  fluid  motion  is  so  largely  based  upon  the  idea  of 
simple  lamellar  flow  that  in  pursuing  the  discussion  we  will  be 
carried  far  away  from  any  consideration  of  the  details  of  actual 


-a^. 


^SSSSEE^^^ 


Fig.  127. 

fluid  motion,  and,  although  many  of  these  details  are  essentially 
erratic,  still  there  are  a  few  details  which  are  definitely  typical. 

The  water  hammer.  —  The  most  striking  phenomenon  that  is 
associated  with  a  varying  state  of  fluid  motion  is  the  effect  pro- 
duced when  an  open  hydrant  is  suddenly  closed ;  the  momen- 
tum of  the  water  in  the  pipe  causes  the  water  to  exert  on  the 


222 


ELEMENTS   OF   MECHANICS. 


suddenly  closed  valve  a  momentary  force  very  much  like  a 
hammer  blow.  This  momentary  force  is  often  excessively  large 
in  value  and  a  valve  which  is  closed  suddenly  should  be  protected 
by  an  air  cushion  as  shown  in  Fig.  127.  The  sharp  rattling 
noise  which  is  occasionally  produced  in  steam  pipes  is  due  to  the 
*'  water  hammer."  A  column  of  condensed  water  is  driven  along 
the  pipe  by  the  steam,  the  cooler  steam  ahead  of  the  column 
condenses,  and  the  column  of  water  hammers  against  the  end  of 
the  pipe  or  against  a  stationary  body  of  water  in  the  pipe. 

The  hydraulic  rmn  consists  of  a  valve  A^  Fig.  128,  arranged 
to  automatically  open  and   close  the  end  of  a  long  pipe    PP. 


FIs.  128. 

When  the  valve  opens  the  water  from  the  dam  starts  to  flow, 
this  flow  lifts  the  valve  A  which  suddenly  closes  the  end  of  the 
pipe  PP^  and  the  momentum  of  the  water  in  PP  generates  a 
momentary  pressure  which  lifts  the  valve  B  and  forces  a  small 
quantity  of  water  to  a  high  storage  tank.  The  valve  A  then 
falls,  and  the  action  is  repeated.  The  automatic  opening  of  the 
valve  A  is  due  to  the  recoil  of  the  water  in  the  pipe  PP  as  fol- 
lows :  At  the  moment  when  the  water  in  PP  is  brought  to  rest 
in  forcing  water  into  the  storage  tank,  the  pressure  at  the  end  of 
PP  is  of  course  still  excessive  and  the  water  near  the  end  of  PP 
is  compressed.  This  compression  then  relieves  itself  by  starting 
a  momentaiy  backward  flow,  or  recoil,   of  the  water  in   PP, 


HYDRAULICS.  223 

and  this  recoil  is  followed  by  a  momentary  decrease  of  pressure 
sufficient  to  allow  the  valve  A  to  drop. 

The  sensitive  flame.  —  When  a  fluid  flows  through  a  fairly 
smooth  walled  pipe,  the  motion  approximates  very  closely  to  a 
simple  flow  if  the  velocity  is  not  excessive  ;  but  when  the  velocity 
is  increased  the  motion  tends  to  become  more  and  more  turbulent 
(full  of  eddies),  and  in  many  cases  there  is  a  fairly  definite  velocity 
at  which  the  motion  suddenly  becomes  very  turbulent.  This  is 
shown  by  watching  the  movement  of  "  sawdust- water  "  through 
a  large  glass  tube.  At  low  velocities  the  particles  of  sawdust 
move  in  fairly  straight  paths,  but  as  the  velocity  of  flow  is  in- 
creased the  particles  begin  to  gyrate  with  considerable  violence 
when  a  certain  velocity  is  reached. 

This  sudden  increase  of  turbulence  is  illustrated  by  the  familiar 
behavior  of  a  gas  flame.  When  the  gas  is  turned  on  more  and 
more  the  flame  remains  fairly  steady  until  the  velocity  of  the 
flowing  gas  reaches  a  certain  critical  value  and  then  the  flame 
suddenly  becomes  rough  and  unsteady.  When  the  flame  is  on 
the  verge  of  becoming  unsteady  it  is  sometimes  very  sensitive ; 
the  least  hissing  noise  causes  it  to  become  turbulent.  An  ex- 
tremely sensitive  flame  may  be  obtained  by  burning  ordinary 
illuminating  gas  from  a  smooth  circular  nozzle  made  by  drawing 
a  glass  tube  down  to  the  desired  size  (about  |^  millimeter  to  i 
millimeter  diameter  of  opening).  Generally,  several  nozzles  must 
be  tried  before  one  is  found  that  is  suited  to  the  gas  pressure  that 
is  available. 

Vortex  rings.  —  When  a  fluid  is  at  rest,  mixing  takes  place 
only  by  the  very  slow  process  of  diffusion,*  and  when  a  fluid  is 
in  turbulent  motion  the  mixing  of  the  different  parts  of  the  fluid 
takes  place  very  rapidly  on  account  of  the  eddies  which  constitute 
the  turbulence.  The  slowness  of  mixing  of  a  smoothly  flowing 
fluid,  however,  is  illustrated  by  the  smooth  gas  flame  and  by  the 
threads  of  smoke  that  rise  from  the  end  of  a  cigar.  Such  a 
stream  of  fluid  flowing  smoothly  through  a  large  body  of  fluid  at 

*  See  chapters  on  heat. 


224  ELEMENTS   OF   MECHANICS. 

rest  tends  always  to  break  up  into  what  are  called  vortex  rings. 
Thus  a  fine  jet  of  colored  water  entering  at  the  top  of  a  large 
vessel  of  clear  water  and  streaming  towards  the  bottom,  breaks 
up  into  rings  which  spread  out  wider  and  wider  as  they  move 
downwards,  each  ring  preserving  its  identity  (not  mixing  with  the 
clear  water).  The  most  interesting  example  of  the  formation  of 
vortex  rings  is  the  familiar  case  of  the  formation  of  smoke  rings 
when  smoke  issues  as  a  moderate  puff  from  an  orifice  into  the 


Fig.  129. 

air.  Of  course  the  smoke  only  serves  to  make  the  rings  visible, 
and  a  candle  can  be  blown  out  by  invisible  vortex  rings  projected 
across  a  large  room  from  an  orifice  in  a  box  by  striking  a  flexible 
diaphragm  which  is  stretched  like  a  drum  head  across  the  open 
side  of  the  box. 

Cyclonic  movements. — When  water  flows  out  of  a  hole  in  the 
bottom  of  a  sink  a  whirlpool  generally  forms  above  the  hole. 
In  order  to  explain  this  rapid  whirling  motion  let  us  consider  a 
weight  W,  Fig.  129,  which  is  twirled  on  a  string  that  winds  up 


HYDRAULICS.  225 

on  a  rod  r  so  as  to  bring  the  weight  continually  nearer  to  the 
center.  The  velocity  of  the  weight  tends  to  remain  unchanged 
in  value  and  therefore  the  number  of  revolutions  per  second  tends 
to  become  greater  and  greater  as  the  weight  approaches  the 
center.*  The  formation  of  a  whirlpool  in  an  emptying  sink  is 
due  to  a  chance  rotating  or  whirling  motion  of  the  water  in  the 
sink  which  may  be  imperceptible,  but  which  becomes  very  greatly 
exaggerated  as  the  water  flows  towards  the  hole  from  all  sides. 

The  rotation  of  the  -earth  on  its  axis  involves  a  slow  motion  of 
turning  of  one's  horizon  about  a  vertical  axis  (except  at  the 
equator).  When  the  warm  air  near  the  earth's  surface  starts  to 
flow  upwards  at  a  given  point,  a  chimney-like  effect  is  produced 
by  the  rising  column  of  warm  air,  the  lower  layers  of  warm  air 
flow  towards  this  ''chimney"  from  all  sides,  and  the  slow  turn- 
ing motion  of  the  horizon  becomes  very  greatly  exaggerated  in  a 
more  or  less  violent  whirl  at  the  "chimney"  which  is  the  center 
of  the  storm.  The  cyclone  is  a  storm  movement  of  this  kind  cov-^^ 
ering  hundreds  of  thousands  of  square  miles  of  territory  with  a 
central  chimney  hundreds  of  miles  in  diameter,  the  tornado  is  a 
storm  movement  of  this  kind  covering  only  a  few  square  miles 
of  territory  with  a  central  chimney  seldom  more  than  a  thousand 
yards  in  diameter.  The  whirling  motion  near  the  center  of  a 
tornado  is  often  excessively  violent. 

124.  Rate  of  discharge  of  a  stream.  —  The  volume  of  water 
which  is  delivered  per  second  by  a  stream  is  called  the  discharge 
rate  of  the  stream.  Thus  the  mean  discharge  rate  of  the  Niagara 
river  is  300,000  cubic  feet  per  second.  The  rate  of  discharge  of 
a  stream  is  equal  to  the  product  of  the  average  velocity,  v,  of  the 
stream  and  its  sectional  area  a.  For  example,  let  PP,  Fig.  130, 
be  the  end  of  a  pipe  out  of  which  water  is  flowing,  and  let  us 
assume  that  the  velocity  of  flow  has  the  same  value  v  over  the 
entire  section  of  the  stream  (lamellar  flow),  then  the  water  which 
flows  out  in  t  seconds  would  make  a  prism  of  which  the  length 

*  Let  the  student  hang  by  his  hands  from  the  end  of  a  long  rope  with  his  feet  held 
as  far  apart  as  possible,  let  a  comrade  set  him  spinning,  and  then  let  the  student  bring 
his  feet  together. 
15 


226-  ELEMENTS    OF   MECHANICS. 

is  vt  and  the  end  area  is  a.  That  is  the  volume  avt  of  water 
flows  out  of  the  pipe  in  /  seconds  so  that  the  discharge  rate  is  av. 
Variation  of  velocity  with  sectional  area  of  a  steady  stream.  — 
Consider  a  simple  flow  of  water  through  a  pipe  as  indicated  by 
the  stream  lines  in  Fig.  125.  Let  a'  and  a"  be  the  cross-sectional 
areas  of  the  stream  at  any  two  points  P'  and  P'\  and  let  v'  and 


Fig.  130. 

v"  be  the  average  velocities  of  the  stream  at  P'  and  P"  respect- 
ively. Then  a'v'  is  the  volume  of  water  which  passes  the  point 
P'  per  second  and  a"v"  is  the  volume  of  water  which  passes 
the  point  P"  per  second  ;  and,  therefore,  since  the  same  amount 
of  water  must  pass  each  point  per  second,  we  have 

a'v'  =  a"v"  (63) 

that  is,  the  product  av  has  the  same  value  all  along  the  pipe, 
so  that  V  is  large  where  a  is  small,  and  v  is  small  where  a  is  large. 
Equation  (63)  applies  to  a  fluid  which  is  approximately  incom- 
pressible ;  that  is,  to  a  liquid,  a  given  amount  of  liquid  having 
always  sensibly  the  same  volume  so  that  the  volume  a'v'  which 
passes  one  point  of  a  steady  stream  per  second  is  equal  to  the 
volume  a"v"  which  passes  any  other  point  of  the  same  stream 
per  second.  In  the  case  of  a  gas,  however,  the  volume  changes 
very  perceptibly  with  pressure  and  equation  (63)  becomes 

a'v'd'  =  a"v"d"  (64) 

where  a'  is  the  sectional  area  of  the  steady  gas  stream  at  one 
place,  v'  is  the  average  velocity  of  the  stream  at  that  place,  d'  is 
the  density  of  the  gas  at  that  place,  and  a" ,  v"  and  d"  are  the 


HYDRAULICS.  22/ 

cross-sectional  area  and  velocity  of  the  stream  and  the  density 
of  the  gas  at  another  part  of  the  stream. 

125.  The  ideal  frictionless  incompressible  fluid.  —  When  a  jet 
of  water  issues  from  a  tank,  there  is  a  certain  relation  between  the 
velocity  of  the  jet  and  the  difference  in  pressure  inside  and  out- 
side of  the  tank.  When  there  are  variations  of  the  velocity 
of  flow  of  water  through  a  pipe  due  to  enlargements  or  contrac- 
tions of  the  pipe  [see  equation  (63)],  the  pressure  decreases 
wherever  the  velocity  increases  and  vice  versa.  These  mutually 
dependent  changes  of  velocity  and  pressure  are  always  compli- 
cated by  friction,  and  by  the  variations  of  the  density  of  the  fluid 
due  to  the  variations  of  pressure;  and  in  order  to  gain  the  simplest 
possible  idea  of  these  mutually  dependent  changes  of  velocity 
and  pressure  the  conception  of  \kiQ  frictionless  incompressible  fluid 
is  very  useful. 

When  the  water  in  a  pail  is  set  in  motion  by  stirring,  it  soon 
comes  to  rest  when  it  is  left  to  itself  A  fluid  which  would  con- 
tinue to  move  indefinitely  after  stirring  would  be  called  a  friction- 
less  fluid. 

When  a  moving  fluid  is  brought  to  rest  by  friction,  the  kinetic 
energy  of  the  moving  fluid  is  converted  into  heat  and  lost.  Such 
a  loss  of  energy  would  not  take  place  in  a  frictionless  fluid,  and 
therefore  the  total  energy  (kinetic  energy  plus  potential  energy) 
of  a  frictionless  fluid  would  be  constant.  This  principle  of  the 
constancy  of  total  energy  is  the  basis  of  the  following  discussion  of 
the  flozv  of  the  ideal  frictionless  fluid.  The  following  discussion 
applies  to  fluids  which  are  also  incompressible.  In  fact,  ordinary 
liquids  are  nearly  incompressible.  The  flow  of  gases  is  discussed 
in  the  chapters  on  heat. 

126.  Energy  of  a  liquid,  {a)  Potential  energy  per  unit  of  volume. 
—  Work  must  be  done  to  pump  a  liquid  into  a  region  under 
pressure,  the  amount  of  work  done  in  pumping  one  unit  of  vol- 
ume of  the  liquid  is  the  potential  energy  per  unit  of  volume  of 
the  liquid  in  the  high  pressure  region,  and  it  is  equal  to  the  pres- 
sure.     That  is  ^,  ^ ^  ^g^^ 


228 


ELEMENTS   OF   MECHANICS. 


In  this  equation  W  and  /  may  both  be  expressed  in  c.g.s.  units 
or  in  the  units  enumerated  at  the  head  of  this  chapter. 

Proof  of  equation  (^5).  —  Let  CC,  Fig.  131,  be  the  cyHnder 
of  a  pump  which  is  used  to  pump  Hquid  into  a  tank  under  a 
pressure  of  /  pounds  per  square  foot,  and  let  the  area  of  the 
piston  be  a  square  feet.  Then  the  force  required  to  move  the 
piston  (ignoring  friction)  is  ap  pounds,  and  the  work  done  in 
moving  the  piston  through  a  distance  of  /  feet  is  apl  foot- 
pounds.    But    al  is  the  volume  of  water  pushed  into  the  tank 


Fig.  131. 

by  the  movement  of  the  piston,  and  therefore,  dividing  apl  by 
al  gives  the  work  in  foot-pounds  required  to  push  one  cubic 
foot  of  water  into  the  tank. 

When  a  stream  of  liquid  moves  in  a  horizontal  plane,  the 
gravity  pull  of  the  earth  does  no  work  on  the  liquid ;  but  when 
a  stream  flows  through  an  inclined  pipe  the  gravity  pull  of  the 
earth  does  work  (positively  or  negatively)  on  the  liquid  and  it  is 
necessary  therefore  in  this  case  to  consider  the  energy  of  altitude 
as  a  part  of  the  potential  energy  of  the  liquid.  In  fact  one  cubic 
foot  of  liquid  of  which  the  weight  is  d  pounds  has  a  potential 
energy  equal  to  hd  foot-pounds  when  it  is  at  a  height  of  Ji  feet 
above  a  chosen  reference  level,  so  that  the  total  potential  energy 
of  the  liquid  per  cubic  foot  is 


W  =p^  hd 


(66) 


HYDRAULICS.  229 

{b)  Kinetic  energy.  —  Let  v  be  the  velocity  in  feet  per  second 
of  a  moving  liquid,  and  let  d  be  the  mass  of  one  cubic  foot  of  the 
liquid  in  pounds  (d  is  the  density  of  the  liquid).  Then  the  kinetic 
energy  of  one  cubic  foot  of  the  liquid  in  foot-pounds  is 

W"  =  —  dv"  (67) 

according  to  equation  (27)  of  chapter  VI. 

127.  Efflux  of  a  liquid  from  a  tank.  —  Consider  a  tank,  Fig. 
132,  containing  a  Hquid  of  which  the  density  is  d  pounds  per 
cubic  foot.     Let    00   be  an 
orifice  from  which  the  liquid  ^s 

issues  as  a  jet  at  a  velocity  .-^1-"^ 

V  feet  per  second  to  be  de-  .-."-".-_"- 

termined.  Let  /  pounds  :i":l:.-I* 
per  square  foot  be  the  pres-  :  -.  ;  1 ;  ;. 
sure  in  the  tank  at  the  level  . :  /  ;  _- : . 
of  the  orifice,  and  let  p'  be  ." "  '  : '. :  : 
the  outside  pressure  (atmo-  -^z-  '=  ^^ 
spheric  pressure).  In  the  =^^-"^'^-^-*— 
tank,  where  the  velocity  of 
the  liquid  is  inappreciable,  the  total  energy  of  the  liquid  per  unit 
of  volume  is  the  potential  energy  p  [equation  (65)].  In  the  jet 
the  total  energy  per  unit  volume  is  p'  -\-  i  j  2g  x  dv^  [equations 
(65)  and  {^yy].  As  a  portion  of  the  liquid  moves  from  the  tank 
into  the  jet  its  total  energy  would  remain  unchanged  if  it  were 
frictionless  so  that  we  would  have 

p=f-\ dv^ 

^      ^        2g 

whence 


Fig.  132, 


W 


5£(^)  (68) 


This  equation  expresses  the  velocity  of  efflux  of  a  frictionless 
incompressible  fluid.  The  effect  of  friction  is  to  decrease  v,  and 
the  effect  of  compressibility  is  to  increase  v.     For  ordinary  liquids 


230 


ELEMENTS   OF   MECHANICS. 


the  effect  of  friction  is  the  greater,  and  equation  (6S)  gives  too 
large  a  value  for  v.  For  gases  the  effect  of  compressibility  is  the 
greater,  and  equation  (68)  gives  too  small  a  value  for  v. 

TorricellVs  theorem.  —  The  velocity  of  efflux  of  a  frictionless 
liquid  is  equal  to  the  velocity  a  body  would  gain  in  falling  freely 
through  the  distance  x  of  Fig.  132.  The  pressure-difference 
p  —  p'  is  equal  to  xd,  according  to  equation  (60)*  of  chapter 
IX,  so  that,  substituting  xd  for  p  —  p'  in  equation  (68),  we 
have 

V  =  V  2gx 

and  this  is  the  velocity  gained  by  a  body  in  falling  through  the 
distance  x,  according  to  Art.  35. 

128.  Diminution  of  pressure  in  a  throat.  —  A  contracted  por- 
tion of  a  pipe  is  called  a  throat.     When  a  fluid  flows  through  a 

A 


pipe  in  which  there  is  a  throat  tJie  velocity  of  the  fluid  in  the  throat 
is  greater  than  it  is  in  the  larger  portions  of  the  pipe,  and  there- 
fore the  pressure  of  the  flidd  in  the  throat  is  less  than  it  is  in  the 
larger  portions  of  the  pipe.  Let  Fig.  1 3  3  represent  a  pipe  with 
a  throat ;  let  a'  be  the  cross-sectional  area  of  the  pipe  at  A,  and 
let  p'  and  v^  be  the  pressure  and  velocity  respectively  of  the  fluid 
at  A  ;  and  let  a" ,  p"  and  v"  be  the  corresponding  quantities  at 
B.     Then,  if  the  fluid  is  incompressible,  we  have 

a'v'  =  a"v"  (i) 

according  to  equation  (63),  and  if  the  fluid  is  also  frictionless  we 
have 


*It  is  to  be  remembered  that  in  equation  (60)  c.g.s.  units  are  used, 
tion  becomes  p^=  xd  for  the  units  used  in  this  chapter. 


This  equa- 


HYDRAULICS. 


231 


^  2g  ^  2g 


(") 


Therefore,  substituting  the  value  of  v^'  from  (i)  in  (ii)  and  solving 
for  p'  —  p'\    we  have 


/-/' 


I      /^'2_^„2 


2^ 


dv' 


(69) 


where  p'  —  p"    is  the  diminution  of  pressure  in  the  throat. 

The  diminution  of  pressure  in  a  throat  is  explained  directly 
from  Newton's  second  law  of  motion  as  follows :  Consider  a 
particle  of  liquid  at  A,  Fig.  133.  This  particle  gains  velocity  as 
it  approaches  B,  and  loses  velocity  again  as  it  approaches  C. 
Therefore  an  unbalanced  force  must  be  pushing  the  particle  for- 
wards as  it  passes  from  A  to  B,  that  is,  the  pressure  behind  the 
particle  is  greater  than  the  pressure  ahead  of  the  particle  ;  and  an 
unbalanced  force  must  be  opposing  the  motion  of  the  particle  as  it 
passes  from  B  to  C,  that  is,  the  pressure  ahead  of  the  particle  is 
greater  than  the  pressure  behind  it. 

Examples.  —  The  diminution  of  pressure  as  a  stream  contracts 
into  a  throat  and  the  rise  of  pressure  as  the  stream  widens  out 
again  are  illustrated  by  several  familiar  devices  as  follows  : 

{a)  The  disk  paradox.  —  Figure  1 34  represents  a  short  piece 
of  tube  T  ending  in  a  flat  disk  DD,  and  dd  is  a  light  metal 
disk  which  is  prevented  from  moving 
sidewise  by  a  pin  which  projects  into  the 
end  of  the  tube  T.  If  one  blows  hard 
through  the  tube  T  the  disk  dd  is  held 
tight  against  DD  because  of  the  low 
pressure  in  the  very  greatly  contracted 
portion  of  the  air  stream  between  the 
disks.  In  fact,  the  pressure  of  the  air 
in  the  region  between  the  disks  is  less 
than  atmospheric  pressure,  and  it  increases  towards  the  edge  of 
the  disks  as  the  velocity  of  the  air  stream  diminishes  (and  the 
sectional  area  of  the  stream  increases). 


Fig.  134. 


232 


ELEMENTS   OF   MECHANICS. 


(b)  The  jet  pump.  —  The  essential  features  of  the  jet  pump  are 
shown  in  Fig.  135.  Water  from  a  fairly  high  pressure  supply  H 
enters  a  narrow  throat,  the  low  pressure  in  the  throat  sucks  water 
from  AA,  and  the  water  from  //,  together  with  the  water  from 
AA^  is  discharged  into  the  reservoir  R.  This  type  of  pump  is 
frequently  used  for  pumping  water  out  of  cellars,  and  it  is  exten- 
sively used  as  an  air  pump  in  chemical  laboratories. 

The  steam  injector  is  a  jet  pump,  and  its  paradoxical  action  in 
pumping  water  into  a  boiler  at  the  same  pressure  as  the  steam  sup- 


m/M/m//m/Mmm 


Fig.  135. 


ply  (or  even  higher),  depends  upon  the  low  density  of  the  steam. 
It  is  evident  from  equation  (68)  that  the  low  density  steam  must 
acquire  a  very  high  velocity  in  flowing  out  of  the  boiler,  whereas 
a  very  much  lower  velocity  suffices  to  carry  the  water  (including 
the  condensed  steam)  back  into  the  boiler. 

(c)  The  volume  of  water  discharged  per  second  from  a  given 
sized  orifice  00^  Fig.  1 36,  is  greatly  increased  by  the  flaring  tube 
AB,  The  rate  of  discharge  of  a  frictionless  fluid  would  depend 
only  upon  the  size  of  the  open  end  B  of  the  tube,  the  contraction 
at  A  would  have  no  effect.     In  the  case  of  an  actual  fluid,  the 


HYDRAULICS. 


22>Z 


effect  of  the  contraction  at  A  is  to  increase  the  friction  consider- 
ably and  thus  reduce  the  discharge  rate  below  what  it  would  be 
if  the  tube  at  A  were  as  large  as  at  B, 

(d)   The  Venturi  water  meter  consists  of  a  throat  inserted  in  a 
water  pipe  through  which  the  water  to  be  measured  flows.     The 


Fig.  136. 

diminution  of  pressure  /'  —  p"  [see  equation  (69)]  is  measured, 
and,  since  the  cross-sectional  areas  a'  and  a"  are  known,  the 
velocity  v'  and  the  rate  of  discharge  a'v'  can  be  calculated 
from  the  measured  value  of  /'  —  /''• 

129.  Reaction  of  a  water  jet.    Force  of  impact  of  a  jet. — Figure. 
137  represents  a  tank  con- 


taining water  at  pressure  p 
(in  excess  of  outside  pres- 
sure). The  tank  has  an 
orifice  of  area  a  and  the 
orifice  is  closed  by  a  plug 
P.  The  force  acting  on 
the  plug  is  equal  to  pa,  and 
the  total  force  pushing  on 
the  side  AA  of  the  tank  is 
equal  to  total  force  pushing 
on  the  side  BB  including 
the  force  acting  on  the  plug. 
Therefore,  it  would  seem 
that   an    unbalanced    force  equal  to  pa   would  push  the  tank 


234 


ELEMENTS   OF   MECHANICS. 


towards  the  left  in  Fig.  137  if  the  plug  were  removed  ;  but  when 
the  plug  is  removed  there  is  a  reduction  of  pressure  in  the 
neighborhood  of  the  orifice  as  indicated  in  Fig.  138,  so  that  the 
unbalanced  force  which  pushes  the  tank  towards  the  left  in  Fig. 
138  is  much  greater  than  pa,  it  is  in  fact  equal  to  2pa  on  the 
following  assumptions,  namely,  {a)  that  the  velocity  of  efflux  is 
that  of  an  ideal  incompressible  fluid,  and  (b)  that  the  jet  issues  as 
a  parallel  stream  of  the  same  size  as  the  orifice. 

It  is  impossible,  however, 
to  show  that  the  reaction  of 
the  jet  is  2pa  by  consider- 
ing the  change  of  pressure 
inside  of  the  tank  due  to 
the  existence  of  the  jet,  but 
the  reaction  can  be  evalu- 
ated in  a  comparatively 
simple  manner  by  consider- 
ing the  force  which  must 
act  on  the  outflowing  water 
to  set  it  in  motion.  In  one 
second  av  cubic  feet  or  avd 
pounds  of  water  flow  out 
of  the  orifice,  and  this  amount  of  water  has  gained  velocity  v. 
To  impart  velocity  v  to  avd  pounds  in  one  second  requires  a  force 
equal  to  adv  x  v  -^  g  pounds- weight,  according  to  equation  (5), 
and  of  course  the  jet  must  push  backwards  upon  the  tank  with  an 
equal  force.  Therefore  the  reaction  of  the  jet  is  adv^jg  pounds- 
weight  ;  but  the  velocity  of  efflux  and  difference  of  pressure 
p\_—  p' — p"  of  equation  (68)]  satisfy  the  equation 


Fig.  138. 


2^ 


dv'^p 


so  that 


-  adv^  =  2pa. 


HYDRAULICS. 


235 


When  a  jet  of  water  strikes  an  obstacle  so  as  to  be  brought  to 
rest,  it  exerts  a  force  equal  to  adv^jg  on  the  obstacle.  If  the 
jet  strikes  a  flat  plate  so  as  to  rebound  in  a  direction  at  right 
angles  to  its  original  velocity,  as  indicated  in  Fig.  139,  then  it 


Fig.  139. 


exerts  the  same  force  as  it  would  exert  if  it  were  brought  to  rest, 
because  it  loses  all  of  its  velocity  in  the  original  direction.  If  the 
jet  strikes  a  curved  plate  as  indicated  in  Fig.  140  so  as  to  rebound 


Fig.  140. 


in  an  opposite  direction  with  unchanged  velocity  (gliding  along 
the  curved  plate  without  friction),  then  it  would  exert  twice  as 
much  force  as  it  would  exert  if  it  were  brought  to  rest,  because  it 


236 


ELEMENTS   OF   MECHANICS. 


would  lose  its  original  velocity  and  gain  an  equal  amount  in  the 
opposite  direction. 

The  Pitot  tube.  —  A  glass  tube  drawn  to  a  moderately  fine 
point  is  placed  in  a  stream  of  water  moving  at  velocity  v^  as 
shown  in  Fig.  141.  In  accordance  with  what  is  stated  above 
concerning  the  reaction  and  impact  of  a  jet,  the  water  in  the  tube 
should  stand  above  the  level  of  the  stream  at  a  height  h  which 
is  approximately  twice  as  great  as  the  height  which  would  cause 
an  efflux  velocity  equal  to  v.  That  is,  the  velocity  of  the  stream 
is  approximately  equal  to  V gh,  according  to  Art.  127.       This 


>_ : 

--^^^--  ->   - 

-"-  -    >  -  -:'®nwi^- 

^ ^ . 

%  •  .   --.  :     •  .-   , 

■  -■.•,•-• 

Fig.  141. 

device  is  called  the  Pitot  tube,  it  is  frequently  used  for  measuring 
the  velocity  of  streams,*  and,  when  so  used,  it  is  usually  arranged 
as  shown  in  Fig.  142,  so  as  to  bring  the  difference  of  level  h  into 
a  convenient  position  for  measurement.  The  tube  A^  Fig.  142, 
has  its  point  directed  against  the  stream,  and  the  tube  B  has  its 
point  directed  at  right  angles  to  the  stream.  By  drawing  the  de- 
vice, Fig.  142,  through  still  water  at  a  known  velocity,  or  by 


*  Other  methods  for  measuring  the  velocity  of  a  stream  are  often  used  in  practice. 
See,  for  example,  Merriman's  Hydraulics. 


HYDRAULICS. 


237 


using  it  to  measure  a  velocity  which  has  been  determined  by- 
other  means,  it  has  been  found  that  its  indications  are  accurate 
to  about  one  per  cent,  when  the  tubes  are  pointed  as  shown  in 
Fig.  142. 

130.  Gauging  streams  — To  gauge  a  stream  is  to  determine 
the  volume  of  water  discharged  by  the  stream  per  second.  This 
determination  depends  upon  the  measurement  of  the  sectional 


Fig.  142. 

area  a  of  the  stream  and  of  the  mean  velocity  v  of  the  stream, 
and  the  discharge  rate  of  the  stream  is  equal  to  av  according  to 
equation  (63). 

Small  streams  are  usually  gauged  by  means  of  an  orifice  in  a 
temporary  dam.*  Let  x  be  the  distance  of  the  center  of  the 
orifice  beneath  the  surface  of  the  water  in  the  dam,  then  the 
velocity  of  efflux  would  be  equal  to  V2gx  if  the  water  were  fric- 
tionless,  and  the  product  of  this  velocity  and  the  area  of  the  ori- 

*  The  arrangement  called  a  iveir  is  a  notch  in  the  top  of  a  temporary  dam,  and  the 
formulas  for  calculating  the  discharge  rate  over  a  weir  may  be  found  in  any  treatise 
on  Hydraulics. 


I 


238  ELEMENTS   OF   MECHANICS. 

fice  a  would  be  the  discharge  rate  if  the  flow  in  the  orifice  were 
lamellar.  Experiments  show  that  the  mean  velocity  of  a  water 
jet  flowing  from  a  sharp  edged  orifice  hke  that  shown  in  Fig.  132 
is  about  0.98  of  the  value,  -|/^,  corresponding  to  ideal  friction- 
less  flow ;  and  experiment  shows  that  the  cross-sectional  area  of 
jet  at  a  short  distance  from  the  orifice  (where  the  flow  becomes 
approximately  lamellar)  is  about  0.62  of  the  area  of  the  orifice, 
provided  the  orifice  has  sharp  edges  and  is  in  the  middle  of  a 
flat  wall.  Therefore  the  rate  of  discharge  from  an  orifice  like  that 
shown  in  Fig.  132  is  approximately  equal  to  0.98  x  0.62 x  a  \/^ 2gx. 

A  large  river  is  gauged  by  determining  the  cross-sectional  area 
of  the  river  and  measuring  the  velocity  of  the  water  at  a  large 
number  of  points  in  the  section  so  as  to  determine  the  average 
velocity.  The  velocity  of  the  current  is  sometimes  measured  by 
means  of  floats,  sometimes  by  means  of  Pitot  tubes,  and  sometimes 
by  means  of  a  so-called  current  meter  which  consists  of  a  rotating 
wheel  like  a  screw  propeller  which  drives  a  speed  counting  device. 
The  current  meter  has  to  be  calibrated  by  observing  its  speeds 
when  it  is  dragged  through  still  water  at  various  known  velocities. 

131.  Fluid  Friction. — The  dragging  forces  which  oppose  the 
motion  of  a  body  through  the  air  or  water,  and  the  dragging  forces 
which  oppose  the  flow  of  fluids  through  pipes  and  channels  are 
due  to  a  type  of  friction  which  is  caWed  ^?itd  fnctioft. 

Friction  of  fluids  in  pipes  and  channels.  —  There  are  two  fairly 
distinct  actions  which  are  involved  in  the  friction  of  fluids  in  pipes 
and  channels,  and,  although  these  two  actions  always  exist  to- 
gether, it  will  be  instructive  to  consider  two  extreme  cases  in 
which  the  two  actions  are  approximately  separated. 

Viscous  Friction.  —  When  a  fluid  flows  through  a  very  small, 
smooth-bore  pipe,  the  loss  of  pressure  is  proportional  to  the  rate 
of  discharge,  or  to  the  mean  velocity  of  flow  of  the  fluid  in  the 
pipe.  This  fact  was  first  established  by  Poiseuille  (1843).  In 
fact,  for  this  case,  we  have 

/=^  (70) 


HYDRAULICS. 


239 


in  which  /  is  the  length  of  the  tube  in  feet,  R  is  the  radius  of  its 
bore  in  fractions  of  a  foot,  Q  is  the  volume  of  liquid  in  fractions 
of  a  cubic  foot  discharged  per  second,  and  ?;  is  a  constant  called 
the  coefficient  of  viscosity  of  the  liquid.  It  is  evident  from  this 
equation  that  the  loss  of  pressure  due  to  viscous  friction  is  very 
small  indeed  when  the  radius  R  of  the  tube  is  moderately  large. 
In  fact,  viscous  friction  is  nearly  always  negligible  under  practical 
conditions.  A  full  discussion  of  equation  (70)  and  a  definition 
of  the  coefficient  of  viscosity  are  given  in  Arts.  132  and  133. 

Eddy  Friction.^  —  Consider  a  series  of  chambers,   ABCD,   Fig. 
143,  communicating  with  each  other  through  narrow  orifices,  and 


Fig.  143. 

let  us  suppose  water  to  flow  through  this  series  of  chambers.  As 
the  water  enters  an  orifice  it  gains  a  certain  amount  of  velocity  v, 
and  decreases  in  pressure  by  the  amount  i/2^x  dv^^  accordmg 
to  Art.  127.  The  velocity  so  gained  is  lost  by  eddy  action  in 
the  next  chamber,  and  when  the  water  flows  through  the  next 
orifice  it  must  gain  velocity  anew  and  suffer  a  corresponding 
drop  in  pressure,  as  before.  It  is  therefore  evident  that  the  drop 
of  pressure  through  the  series  of  chambers,  ABCD,  is  proportional 
to  the  square  of  the  rate  of  discharge.  This  law  of  eddy  friction  is 
verified  by  experiment  for  a  series  of  chambers  as  shown  in  Fig. 
143,  where  the  eddies  are  definitely  localized.  In  an  ordinary 
pipe,  however,  there  is  a  tendency  for  the  eddy  movements  to  be- 
come finer  grained,  as  it  were,  with  increasing  velocity ;  that  is, 
with  increased  velocity  a  given  particle  of  fluid  acquires  velocity 
and  loses  it  again  an  increased  number  of  times  in  traveling  a 

*  A  fluid  entirely  devoid  of  viscosity  would  not  form  eddies,  so  that  all  fluid  friction 
is  due  to  viscosity  directly  or  indirectly. 


240   •  ELEMENTS   OF   MECHANICS. 

given  distance.  The  consequence  of  this  fact  is  that  the  loss  of 
pressure  due  to  eddy  friction  increases  more  rapidly  than  in  pro- 
portion to  the  square  of  the  rate  of  discharge. 

In  all  ordinary  cases  of  the  flow  of  fluids  through  pipes  and 
channels,  eddy  friction  is  very  much  larger  than  viscous  friction, 
and  the  practical  formula  for  calculating  the  loss  of  pressure  due 
to  the  flow  of  a  fluid  through  a  given  length  of  pipe  of  a  given 
size  is  based  upon  the  assumption  that  the  loss  of  pressure  is  pro- 
portional to  the  density  of  the  fluid  and  to  the  square  of  the  rate 
of  discharge,  or  indeed,  to  the  square  of  the  velocity  of  the  fluid 
if  the  pipe  is  of  uniform  size. 

The  meaning  *  of  the  practical  formula  may  be  made  clear  by 
the  following  argument :  The  flow  of  a  fluid  over  a  surface  such 
as  the  interior  walls  of  a  pipe  is  retarded  by  a  force  which  is 
approximately  proportional  to  the  area  of  the  surface,  to  the 
density  of  the  fluid  and  to  the  square  of  the  velocity  at  which 
the  fluid  is  flowing.     Therefore  we  may  write 

F=  kadiP" 

in  which  a  is  the  area  of  the  surface  in  square  feet,  d  is  the  density 
of  the  fluid  in  pounds  per  cubic  foot,  v  is  the  velocity  of  flow  in 
feet  per  second,  and  i^is  the  retarding  force  in  pounds-weight. 
The  quantity  k  is  called  the  coefficient  of  friction  of  the  moving 
fluid  against  the  walls  of  the  pipe.  It  depends  greatly  upon  the 
degree  of  roughness  of  the  walls  ;  and,  for  a  given  degree  of 
roughness,  it  is  not  strictly  constant,  that  is  to  say,  the  friction  is 
not  exactly  proportional  to  the  square  of  the  velocity. 

Consider  a  pipe  of  which  the  length  is  /  feet,  and  the  inside 
diameter  D  feet.  The  total  area  of  interior  walls  of  this  pipe  is 
irDl  square  feet,  so  that,  using  ttDI  for  a  in  the  above  equation, 
we  have  F=  kirDldv^  for  the  total  retarding  force  acting  on  a 
fluid  of  density  d  flowing  through  the  pipe  at  velocity  v.  This 
retarding  force  is  equal  to  the  difference  of  pressure  at  the  two 

■'^  The  formula  is  not  rational,  it  is  empirical,  and  the  only  thing  to  be  done  in  con- 
nection with  it  is  to  exhibit  its  meaning  clearly. 


HYDRAULICS.  24 1 

ends  of  the  pipe  multiplied  by  the  sectional  area  of  the  bore  of 
the  pipe.  Therefore,  using  /  for  the  loss  of  pressure  due  to  fric- 
tion in  the  pipe,  we  have 

^     4 
whence 

Akldv^  ,     ^ 

/ — ^-  (71) 

The  presence  of  elbows  and  valves  causes  excessive  eddies  and 
therefore  an  excessive  loss  of  pressure  by  friction.  Methods  for 
estimating  the  effects  of  elbows  and  valves  are  given  in  standard 
works  on  hydraulics. 

Example  i.  An  iron  pipe  one  foot  in  diameter  and  10,000 
feet  long  discharges  4.25  cubic  feet  per  second  of  water  when  the 
pressure  at  one  end  is  6,000  pounds  per  square  foot  greater  than 
the  pressure  at  the  other  end.  A  discharge  of  4.25  cubic  feet 
per  second  corresponds  to  a  velocity  of  5.41  feet  per  second  in 
the  pipe  (=  v).  The  density  of  the  water  is  62J  pounds  per 
cubic  foot  (=  ^).  Substituting  these  values  in  equation  (71) 
and  we  find  for  the  coefficient  k  the  value  0.0000731. 

Example  2.  Compressed  air  at  a  mean  pressure  of  5.42  atmos- 
pheres (density  0.406  pounds  per  cubic  foot)  is  forced  through 
15,000  feet  of  pipe  8  inches  inside  diameter  at  a  velocity  of  19.32 
feet  per  second  with  a  difference  in  pressure  of  5.29  pounds  per 
square  inch  between  the  two  ends  of  the  pipe.  Reducing  these 
data  to  the  units  employed  in  this  chapter  and  substituting  in 
equation  (71),  we  have  for  the  coefficient  k  the  value  0.0000534. 

These  two  examples  indicate  the  method  of  determining  the 
approximate  value  of  the  coefficient  k  under  given  conditions. 
When  the  value  k  has  been  so  determined,  equation  (71)  may  be 
used  for  determining  the  amount  of  pressure  required  to  force  a 
given  fluid  through  a  given  pipe  at  given  velocity,  or  it  may  be 
used  to  determine  the  velocity  at  which  a  given  pressure  will 
force  a  given  fluid  through  a  given  pipe.  The  value  of  the 
coefficient  k  is  always  open  to  question,  inasmuch  as  it  depends 
16 


242.  ELEMENTS   OF   MECHANICS. 

greatly  upon  the  degree  of  roughness  of  the  walls  of  the  pipe, 
and,  in  fact,  it  depends  upon  the  size  of  the  pipe  and  upon  the 
velocity  of  the  fluid ;  it  varies  from  about  0.00016  for  new  iron 
pipes  o.  I  foot  inside  diameter,  to  about  0.00005  ^^^  ^^w  iron  pipes 
six  feet  in  diameter,  for  velocities  of  three  or  four  feet  per  second. 

Resistance  of  ships.  —  The  forces  which  oppose  the  motion  of 
a  ship  depend  upon  two  distinct  actions  which  are  called  ivave 
friction  and  eddy  friction  respectively  ;  viscous  friction,  in  the 
sense  in  which  the  term  is  defined  above,  is  entirely  negligible. 

Wave  friction.  — The  motion  of  a  boat  causes  a  heaping  up  of 
the  water  at  the  bow  and  the  production  of  a  hollow  at  the  stern. 
The  former  exerts  a  backward  pressure  on  the  boat,  and  the  lat- 
ter causes  the  forward  pressure  on  the  stern  to  be  less  than  it 
would  be  if  the  water  closed  in  at  the  stern  without  changing  its 
level.  These  changes  of  level  at  the  bow  and  stern  give  rise  to 
waves,  and  the  work  that  is  done  in  driving  the  boat  forward  in 
opposition  to  the  forces  due  to  changes  of  level  at  the  bow  and 
stern  is  carried  away  or  ''  radiated  "  by  these  waves.  Wave  fric- 
tion frequently  amounts  to  more  than  half  of  the  total  friction  of 
a  boat  at  high  speeds. 

Eddy  friction.  —  Anyone  who  has  watched  the  water  over  the 
side  of  a  rapidly  moving  steamer  will  have  noticed  next  to  the 
hull  a  very  turbulent  layer  of  water  several  inches  in  thickness. 
This  turbulence  involves,  of  course,  a  loss  of  energy  and  there- 
fore a  frictional  drag  upon  the  boat.  This  frictional  drag  is  gen- 
erally  called  skin  friction.  There  is  also,  near  the  bow  and 
stern,  regions  of  turbulence  which  extend  to  some  distance  from 
the  boat,  especially  if  the  boat  is  faulty  in  shape,  and  the  frictional 
drag  associated  with  this  extended  turbulence  is  usually  called 
eddy  friction  by  marine  engineers. 

The  friction  which  opposes  the  motion  of  a  boat  is  always  ex- 
pressed in  terms  of  the  actual  force  with  which  the  water  drags 
backward  on  the  boat,  whereas  the  frictional  opposition  to  the  flow 
of  a  fluid  through  a  pipe  is  usually  specified  in  terms  of  the  differ- 
ence of  pressure  at  the  two  ends  of  the  pipe  due  to  the  friction. 


HYDRAULICS.  243 

132.  Definition  of  the  coefficient  of  viscosity  of  a  fluid.  —  Consider  a  thin 
layer  of  fluid  of  thickness  x  between  two  flat  plates  A  A  and  BB  as  shown  in  Fig. 
144,  and  suppose  that  the  plate  A^  is  moving  at  velocity  z>  as  indicated  by  the  ar- 
rows.    If  the  fluid  between  the  plates   AB   were  a  viscous  liquid  like  syrup,  it  is 


B 

Fig.  144. 


evident  that  a  very  considerable  force  would  have  to  be  exerted  upon  the  plate  A  A 
to  keep  it  in  motion  ;  in  fact  any  fluid  whatever,  whether  liquid  or  gas,  is  more  or 
less  like  syrup  in  this  respect,  and  the  force  F  with  which  the  motion  of  the  plate  is 
opposed  by  the  fluid  is  proportional  to  its  area  «,  to  its  velocity  v  and  inversely  pro- 
portional to  the  distance  x  between  the  plates.     That  is 


F=1--^  (72) 


in  which  the  proportionality  factor  t]  is  called  the  coefficient  of  viscosity  of  the  fluid. 

Examples.  The  coefficient  of  viscosity  of  water  at  ordinary  room  temperature  is 
0.0000215  and  the  coefficient  of  viscosity  of  good  machine  oil  is  about  0.00085,  ^>  ^i 
V  and  X  being  expressed  in  terms  of  the  units  specified  at  the  beginning  of  this  chapter. 
It  may  seem,  therefore,  that  water  would  be  a  better  lubricant  than  the  oil,  but  a  layer 
of  water  would  quickly  flow  out  from  between  a  shaft  and  a  bearing  surface,  whereas 
a  rotating  shaft  continually  carries  a  fresh  supply  of  a  viscous  liquid  like  oil  into  the 
space  between  the  shaft  and  the  bearing  surface. 

133.  Flow  of  a  viscous  liquid  through  a  small  smooth-bore  tube.  —  Let  R 
be  the  radius  of  the  bore  of  the  tube,  /  the  length  of  the  tube,  /  the  difference  of  pres- 
sure of  the  liquid  at  the  ends  of  the  tube,  and  v  the  velocity  of  the  liquid  at  a  point 
distant  r  from  the  axis  of  the  tube.  Consider  a  cylindrical  portion  of  the  moving 
liquid  of  radius  r  and  coaxial  with  the  tube.  The  surface  of  this  cylindrical  portion  of 
liquid  moves  as  a  solid  rod  through  the  tube  at  velocity  v.  Similarly,  the  cylindrical 
surface  of  radius  r  -|-  Ar  moves  through  the  tube  as  a  hollow  shell  2X  velocity  v  —  Az/. 
The  layer  of  liquid  between  this  rod  and  shell  is  under  the  same  conditions  of  motion 
as  the  layer  of  liquid  between  the  plates  AA  and  BB  in  Fig.  144.  Therefore, 
writing  Az/  for  v  in  equation  (72),  writing  Ar  for  jr,  and  writing  27rr/  for  a  we 
have 

^  1-Krl  •  Az; 

F=irj 

Ar 

where  F  is  the  force  which  must  act  on  the  end  of  the  rod  to  overcome  the  viscous 
drag  ;  but  this  force  is  equal  to  the  area  of  the  end  of  the  rod  multiplied  by  p,  so  that 


244 


ELEMENTS   OF   MECHANICS. 


or 


whence* 


Tcr'^p  =  TJ 


dv 


27rr/  •  Av 


dr       2T]l 


a  constant 


but  when   r  ^=  R,    z/  =  o,    so  that  the  constant  of  integration  is  equal  to   —pR^\\nl 
and  therefore 

_  pr'^      pR^ 


(i) 


The  velocity  at  each  part  of  the  tube  is  thus  determined.     To  find  the  volume  V 
of  fluid  discharged  in  time  r,  consider  a  section  of  the  tube,  Fig.  145.     The  velocity 

over  all  the  area,  27rrAr,  of  the  dotted  annulus, 
is  Vy  so  that  the  volume  A  F,  flowing  across  this 
annulus  in  time  r,  is  A  K=  27rrA;>'  -v-t.  Sub- 
stituting V  from  (i),  we  have 

d  V=  -At-  r^dr ^— —  rdr 

2lTj 
TTpR^T     PR 


T  nR 


2/r/     Jo 


Fig.  145. 


Problems. 

156.  Find  the  mean  velocity  at  which  water  must  flow  in  a 
canal  20  feet  wide  and  6  feet  deep,  in  order  that  the  rate  of  dis- 
charge may  be  500  cubic  feet  per  second. 

How  many  acres  of  storage  basin  would  be  required  to  store 
an  amount  of  water  sufficient  to  maintain  this  flow  of  water  for 
24  hours,  the  average  depth  of  the  water  in  the  storage  basin  to 
be  I  o  feet  ? 

157.  The  density  of  gas  in  a  steady  stream  is  0.20  pound  per 
cubic  foot  at  one  point  and  0.35  pound  per  cubic  foot  at  another 
point.  The  section  of  the  stream  is  0.25  square  inch  at  the  first 
point  and  0.56  square  inch  at  the  other  point.  Find  the  ratio 
of  the  velocities  of  the  stream  at  the  two  points. 


HYDRAULICS. 


245 


158.  How  much  work  in  foot-pounds  is  required  to  pump 
10,000  cubic  inches  of  water  into  a  reservoir  in  which  the  pres- 
sure stands  at  the  constant  value  of  1 50  pounds  per  square  inch 
above  atmospheric  pressure  ? 

159.  The  velocity  of  a  water  jet  is  200  feet  per  second,  what 
is  the  kinetic  energy  of  the  water  in  foot-pounds  per  cubic  inch  ? 
One  cubic  inch  of  water  weighs  0.0376  pound. 

160.  Calculate  the  velocity  of  efflux  of  kerosene  from  a  vessel 
in  which  the  pressure  is  52  pounds  per  square  inch  above  atmos- 
phere pressure.  The  density  of  kerosene  is  0.03  pound  per  cubic 
inch. 

161.  Water  flows  in  a  12-inch  main  at  a  velocity  of  4  feet  per 
second  and  encounters  a  partly  closed  valve  through  which  the 
section  of  the  stream  is  reduced  to  0.36  square  foot.  Calculate 
the  loss  of  pressure  at  the  valve  due  to  friction. 

Note.  —  As  the  water  enters  the  narrow  passageway  in  the  valve,  its  velocity  in- 
creases by  a  definite  amount,  and  its  pressure  falls  off"  accordingly,  as  explained  in 
Arts.  127  and  128.  As  the  water  issues  from  the  narrow  passageway,  it  retains  its 
velocity  as  a  jet  flowing  through  the  surrounding  water,  so  that  its  pressure  does  not 
rise  again,  and  the  excess  of  velocity  is  then  destroyed  bj  eddy  action.  Therefore 
the  loss  of  pressure  through  the  valve  is  approximately  equal  to  the  drop  of  pressure 
due  to  the  increased  velocity  of  the  water  as  it  enters  the  narrow  passage. 

When  a  portion  of  a  moving  fluid  meets  with  another  portion  which  is  moving  at 
a  less  velocity,  the  excess  of  velocity  is  lost  in  eddy  motion  and  we  have  what  is 
frequently  called  a  "  shock." 

162.  A  street  water-main  7  inches  inside  diameter  has  a  throat 
3  inches  in  diameter  inserted  in  it. 
The  flow  of  water  through  the 
pipe  is  I  J^  cubic  feet  per  second 
and  the  pressure  in  the  7 -inch 
pipe  is  90  pounds  per  square 
inch.  What  is  the  pressure  in 
the  throat  in  pounds  per  square 
inch,  ignoring  friction  ? 

163.  The  difference  in  level,  h,  ^^^'  ''^^^' 

Fig.  1 63/*,  is  observed  to  be  6  inches.     Calculate  the  rate  of  dis- 
charge of  water  through  the  pipe  in  cubic  feet  per  second. 


246  . 


ELEMENTS   OF   MECHANICS. 


Note. — The  specific  gravity  of  mercury  is  13.6,  and  the  tube  AB  is  entirely 
filled  with  water  above  the  surface  of  the  mercury. 

164.  A  pair  of  Pitot  tubes  is  placed  in  the  stream  of  air  issuing 
from  a  fan  blower,  as  shown  in  Fig.  164/,  and  the  difference  in 

level,  hy  is  observed  to  be 
;  '.  ':■  •;  • '/ ,  v  •  •'    .  • ..;  ^.  *  r  :.      ••  2  J^  inches,  the  tubes  being 

filled  with  water.  Calcu- 
late the  velocity  of  the  air 
stream. 

iVi?/<r.  —  In  this  problem  ignore 
the  compressibility  of  the  air,  and 
assume  its  specific  gravity  to  be 
0.08  pound  per  cubic  foot. 

165.  A  temporary  dam 
made  of  thin  boards  has  a 
circular  hole  i  foot  in  di- 
ameter cut  through  it,  and 
the  water  rises  to  a  height 
of  I  }4  feet  above  the  cen- 
ter of  the  hole.  Calculate  the  discharge  rate  of  the  stream  in 
cubic  feet  per  second. 

166.  A  supply  of  10  cubic  feet  per  second  of  water  is  to  be 
brought  from  a  reservoir  to  a  center  of  distribution  in  a  small 
city.  The  height  of  the  surface  of  the  reservoir  above  the  point 
of  distribution  in  the  city  is  350  feet  and  it  is  desired  to  have  an 
available  head  of  150  feet  at  the  center  of  distribution.  Required 
the  size  of  pipe  that  is  necessary  to  deliver  the  water,  the  length 
of  pipe  being  1 6,000  feet. 

167.  A  water  tank  is  installed  for  the  protection  of  a  factory 
against  fire.  The  water  level  in  the  tank  is  75  feet  above  a  cer- 
tain fire  hydrant  in  the  building.  The  pipe  leading  from  the  tank 
to  the  fire  hydrant  consists  of  150  feet  of  4-inch  pipe  in  which 
there  is  one  Pratt  and  Cady  check  valve,  and  four  short-turn 
ells  ;  and  200  feet  of  3 -inch  pipe  in  which  there  are  three  short- 
turn  ells.     Find  the  number  of  gallons  of  water  per  second  that 


Fig.  164A 


HYDRAULICS.  247 

can  be  delivered  at  the  hydrant  allowing  a  loss  of  head  of  25  feet 
in  the  pipe. 

Note.  —  A  four-inch  Pratt  and  Cady  check  valve  has  a  resistance  equivalent  to  25 
feet  of  four-inch  pipe,  and  short-turn  ells  each  have  a  resistance  equivalent  to  4  feet 
of  pipe  of  the  same  size. 


CHAPTER   XI. 
WAVE   MOTION   AND   OSCILLATORY   MOTION. 

134.  Wave  motion  as  a  basis  of  certain  branches  of  physical 
theory.  Scope  of  this  chapter.  —  Many  of  the  fundamental  prin- 
ciples of  mechanics,  such  as  Newton's  laws  of  motion  and  the 
principle  of  the  conservation  of  energy,  are  used  throughout  the 
whole  range  of  the  physical  sciences.  Indeed,  the  principle  of 
the  conservation  of  energy  is  so  widely  used  in  the  study  of 
physics  and  chemistry  that  its  purely  mechanical  origin  *  is  gen- 
erally lost  sight  of.  There  "is,  however,  another  aspect  in  which 
mechanics  is  utilized  in  connection  with  general  physics,  inasmuch 
as  many  physical  and  chemical  theories  are  essentially  mechanical. 
Thus,  the  molecular  theory  in  chemistry  is  as  truly  a  mechanism 
as  a  locomotive,  except  that  a  locomotive  must  be  actually  con- 
structed to  be  used,  whereas  the  molecular  theory  need  only  be 
imagined,  t 

The  wave  theory.  —  That  group  of  ideas  which  relates  to  the 
kind  of  mechanical  action  called  wave  motion,  is  perhaps  more 
useful  in  general  physics  than  any  other  group  of  mechanical 
ideas,  not  excepting  even  the  group  of  mechanical  ideas  which  is 
called  the  molecular  theory.  Nearly  every  phenomenon  of  sound 
and  light,  and  nearly  all  of  the  phenomena  of  oscillatory  motion 
become  intelligible  in  terms  of  the  ideas  of  wave  motion. 

Complexity  of  water  waves.  —  In  undertaking  to  establish  the 
more  important  ideas  of  wave  motion  we  are  confronted  with  a 
serious  difficulty,  namely,  that  water  waves,  the  only  kind  of 

*  See  Art.  59. 

■j-  The  student  should  guard  against  the  idea  that  it  is  easier  to  imagine  the  molecular 
theory  than  it  would  be  to  build  a  locomotive,  for  this  is  by  no  means  the  case.  A 
million  men,  perhaps,  can  be  found  capable  of  building  a  locomotive  that  will  work, 
where  one  can  be  found  who  can  imagine  the  molecular  theory  so  that  it  will  work  ; 
the  final  test  is  the  same  in  both  cases  :  will  the  thing  work  ? 

248 


WAVE   MOTION   AND   OSCILLATORY   MOTION.  249 

waves  with  which  every  one  is  familiar,  are  excessively  compli- 
cated ;  invisible  sound  waves  in  the  air  and  the  even  more  in- 
tangible light  waves  in  the  ether,  in  their  more  important  aspects, 
at  least,  are  extremely  simple  in  comparison.  The  wave  theory, 
however,  originated  in  the  application,  to  sound  and  light,  of  the 
ideas  which  grew  out  of  a  familiarity  with  the  behavior  of  water 
waves,  and,  in  attempting  to  establish  the  wave  theory,  one  is 
obliged  to  base  it  upon  the  familiar  phenomena  of  wave  motion 
as  exemplified  by  water  waves. 

Wave  media.  —  The  material  or  substance  through  which  a 
wave  passes  is  called  a  medium.  Thus,  the  air  is  the  medium 
which  transmits  sound  waves,  and  the  ether  is  the  medium  which 
transmits  light  waves.  A  wire  (or  a  rod)  may  be  spoken  of  as  a 
medium  when  one  is  concerned  with  the  transmission  of  waves 
along  the  wire  (or  the  rod).  During  the  passage  of  the  wave  the 
medium  always  moves  to  some  extent,  but  the  velocity  with  which 
the  medium  actually  moves  is  generally  very  much  less  than  the 
velocity  of  progression  of  the  wave.  *  Thus,  when  the  end  of  a  long 
rope  is  moved  rapidly  to  and  fro  sidewise,  waves  travel  along  the 
rope  and  each  point  of  the  rope  oscillates  as  the  waves  pass  by.  In 
some  cases  the  medium  is  left  permanently  displaced  after  the 
passage  of  the  wave ;  in  other  cases,  the  medium  returns  to  its 
initial  position  after  the  passage  of  the  wave. 

Wave  pulses  and  wave  trains.  —  When  a  stone  is  pitched  into 
a  pond  a  wave  emanates  from  the  place  where  the  stone  strikes. 
When  a  long  stretched  wire  is  struck  sharply,  with  a  hammer,  a 
single  wave  (a  bend  in  the  wire)  travels  along  the  wire  in  both 
directions  from  the  point  where  the  wire  is  struck.  When  a  long 
steel  rod  is  struck  on  the  end  with  a  hammer,  a  single  wave  (an 
endwise  compression  of  the  rod)  travels  along  the  rod.  When 
an  explosion  takes  place  in  the  air,  the  firing  of  a  gun,  for  example, 
a  single  wave  (a  compression  of  the  air)  travels  outwards  from  the 

*  The  mathematical  theory  of  wave  motion  is  developed  on  the  assumption  that 
the  actual  velocity  v  of  the  medium  is  very  small  in  comparison  with  the  velocity  of 
wave  progression   V. 


250  •  ELEMENTS   OF   MECHANICS. 

explosion.  Such  isolated  waves  are  called  wave  pulses.  When  a 
disturbance  at  a  point  in  a  medium  is  repeated  in  equal  intervals 
of  time,  the  disturbance  is  said  to  be  periodic.  Such  a  disturbance 
sends  out  a  succession  of  similar  waves  constituting  what  is  called 
a  wave  train. 

The  wave  pulse  involves  all  *  of  the  important  mechanical 
actions  of  wave  motion,  and  the  mechanics  of  wave  motion  can 
be  developed  in  the  simplest  possible  manner  by  considering  the 
behavior  of  wave  pulses. 

The  applications  of  the  wave  theory  to  sound  and  light  depend 
very  largely  upon  a  consideration  of  wave  trains,  but  the  theory 
of  wave  trains  does  not  have  any  important  applications  in  me- 
chanics. This  chapter  is  limited  therefore  to  the  study  of  wave 
pulses. 

Wave  shape.  —  A  term  which  is  frequently  used  in  the  discus- 
sion of  wave  motion  is  the  term  wave  shape,  and  the  meaning  of 
this  term  may  be  best  explained  by  considering  wave  motion  along 
a  stretched  wire.  Let  us  consider  first  an  entirely  general  case 
as  follows  :  Let   AB^    Fig.  146,  be  a  wire  under  a  tension  of  T 


Vfiie  B 


Fig.  146. 

pounds-weight,  and  suppose  that  each  foot  of  the  wire  weighs  in 
pounds.  Imagine  the  wire  to  be  drawn  at  a  velocity  of  V  feet 
per  second  through  a  crooked  tube,  WW,  and  let  it  be  assumed 
that  the  wire  slides  through  the  tube  without  friction.     Then  the 

*The  mechanical  action  which  serves  as  a  basis  for  the  theory  of  the  dispersion  of 
light  depends  upon  the  effect  a  wave  train  has  upon  the  particles  of  the  medium  where 
these  particles  have  a  tendency  to  oscillate  at  a  definite  frequency,  but  a  discussion  of 
this  effect  is  beyond  the  scope  of  an  elementary  text. 


WAVE   MOTION  AND  OSCILLATORY   MOTION.  251 

wire  will  not  exert  any  force  against  the  sides  of  the  tube,  if  the 
velocity  F  satisfies  the  equation 


1  r-n 

=  aI"^  (73) 


\  m 

and  at  this  velocity  the  moving  wire  would  therefore  retain  its 
crooked  (stationary)  shape  if  the  tube  could  be  removed.  This 
tendency  of  a  bend,  once  established  in  a  rapidly  moving  flexible 
wire  or  chain,  to  persist  is  strikingly  illustrated  by  the  behavior 
of  the  loose  chain  of  a  differential  pulley  *  when  the  pulley  is 
rapidly  lowered  and  the  loose  chain  set  into  rapid  motion,  and  a 
series  of  stationary  bends  is  often  seen  in  a  rapidly  moving  belt. 

The  absence  of  force  action  between  tube  and  wire  in  Fig.  146 
may  be  shown  as  follows  :  Consider  any  point,  /,  of  the  tube. 
The  portion  of  the  tube  in  the  immediate  neighborhood  of  this 
point  is  necessarily  a  portion  of  a  circle  of  a  certain  radius,  r. 
Therefore,  if  the  wire  were  stationary,  its  tension  would  produce 
a  force  against  the  side,  d,  of  the  tube,  and  this  force  would  be 
equal  to  Tjr  pounds  per  foot  of  length  of  wire,  according  to  Art. 
40.  But  to  constrain  the  particles  of  the  wire  to  the  small  circu- 
lar arc  at  /,  an  unbalanced  force  equal  to  m  V^jirg)  pounds- 
weight  per  foot  must  act  on  the  wire,  pulling  it  towards  the 
side,  d^  of  the  tube,  according  to  Art.  38.  Therefore,  when 
T/r  =  m  V^jirg)  or  when  V=  VgTjin,  the  side  force  due  to 
the  tension  of  the  wire  is  just  sufficient  to  constrain  the  particles 
of  the  wire  to  the  curved  path  at  /,  whatever  the  curvature  at 
that  point  may  be,  and  no  force  need  act  upon  either  side  of  the 
tube. 

We  may  imagine  everything  in  Fig.  146  (moving  wire  and 
tube)  to  be  set  moving  to  the  right  at  a  velocity  equal  and  oppo- 
site 19  the  velocity  at  which  the  wire  is  moving  to  the  left.  The 
result  would  be  that  the  wire  would  be  stationary  and  the  tube 
would  move  to  the  left  at  velocity  V,  and,  of  course,  the  wire 
would  pass  through  the  tube  without  exerting  any  force  upon  the 

*  See  figure  illustrating  problem  84  on  p.  126. 


252  ELEMENTS   OF   MECHANICS. 

sides  of  the  tube,  as  before,  so  that  the  bend,  IVW,  would  con- 
tinue to  move  along  the  wire  without  changing  its  shape,  even  if 
the  tube  were  non-existent.  Such  a  moving  bend  constitutes  a 
wave,  and  the  only  motion  of  a  given  point  of  the  wire  during  the 
passage  of  the  wave  would  be  its  sidewise  motion.  Tke  term 
wave  shape  refers  to  the  distribution  of  the  velocity  of  the  7nedium 
{sidewise  velocity  of  the  wire  in  Fig.  146)  in  a  wave.  This  mat- 
ter may  be  made  clear  by  the  following  example.  Imagine  the 
straight  tube,  WW,  Fig.  147,  to  slide  along  the  wire,  AB,  at 
velocity  F,  thus  producing  a  wave.     The  wire  at  each  point  in 


wire 


-it 


% 


Fig.  147. 

the  tube  is  moving  sidewise  at  a  constant  velocity  as  indicated  by 
the  small  arrows,  and  the  portion  of  the  wire  in  the  tube  is  uni- 
formly stretched.  The  uniform  stretch  of  the  wire  in  the  tube  in 
Fig.  147  is  evident  when  we  consider  first  that  the  horizontal  com- 
ponent of  the  tension  of  the  wire  in  the  tube  is  equal  to  the  ten- 
sion, T,  of  the  portions  of  the  wire  beyond  the  tube,  so  that  the 
tension  of  the  wire  in  the  tube  is  greater  than  T,  and  second  that 
the  tube  is  straight  so  that  the  tension  of  the  wire  in  the  tube  is 
uniform. 

In  discussing  waves  it  is  convenient  to  draw  the  line,  AB, 
Fig.  148,  in  the  direction  of  progression  of  the  wave  and  to  rep- 
resent the  actual  velocity  of  the  medium  at  each  point  in  the 
wave  by  an  ordinate,  y,  as  shown ;  or  to  represent  the  actual 
stretch  (or  compression)  of  the  medium  at  each  point  by  the 
ordinate,  jF.  Thus,  the  wave  shown  in  Fig.  147  would  be  rep- 
resented by  the  rectangle  in   Fig.    148,   inasmuch   as   the   side 


WAVE   MOTION   AND    OSCILLATORY   MOTION. 


253 


velocity  of  the  wire  and  the  stretch  of  the  wire  are  everywhere 
constant  in  value  between  WW.  The  wave  represented  in  Fig. 
147  is  therefore  called  a  rectangular  wave,  or  a  rectangular  wave 
pulse.  The  entire  discussion  of  wave  pulses  in  Arts,  ij^  to  ij8 
refers  to  rectangular  wave  pulses^  because  rectangular  wave  pulses 
are  the  simplest  to  describe. 

Pure  and  impure  waves. "^  —  In  certain  cases  a  wave  pulse  re- 
tains its  shape  as  it  travels  through  a  medium.  Thus,  for  ex- 
ample, a  bend  travels  along  a  stretched  flexible  string  or  wire 
with  unchanging  shape.  Such  a  wave  is  called  a  pure  wave.  In 
other  cases,  a  wave  spreads  out  more  and  more  as  it  travels 


vrire                      T 

— >-r 

wire 

i 

axis  of 

A 

1 

JB 

progression 


Fig.  148. 


through  a  medium.  Thus,  for  example,  a  sharply  defined  water 
wave  in  a  canal  spreads  out  over  a  greater  and  greater  length  of 
the  canal  and  becomes  less  sharply  defined  as  it  moves  along. 
Such  a  wave  is  called  an  impure  wave.  Waves  are  generally 
rendered  impure  by  friction  or  by  the  imperfect  elasticity  of  a 
medium.     This  matter  is  discussed  very  briefly  in  Art.  135. 

The  distinction  between  pure  and  impure  waves  is  very  impor- 
tant in  telephone  engineering.      The  transmission  of  articulate 

*  The  best  discussion  of  the  difference  between  pure  and  impure  waves  is  that  which 
is  given  by  Heaviside  in  his  Electro- Magnetic  Theory,  Vol.  I,  pages  307  to  466.  As 
an  introduction  to  this  discussion  by  Heaviside,  the  student  should  read  an  article  on 
Electric  Waves  and  the  Behavior  of  Long  Distance  Telephone  Lines,  by  W.  S.  Frank- 
lin, Journal  of  the  Franklin  Institute,  July,  1905. 


254  ELEMENTS   OF   MECHANICS. 

Speech  by  the  telephone  depends  upon  the  passage  of  sharply 
defined  electric  waves  along  the  telephone  wires.  If  these  waves 
remain  pure,  every  detail  of  shape  is  retained  as  the  wave  moves 
along,  whereas,  if  the  waves  become  impure,  each  part  of  every 
wave  begins  to  spread  over  the  adjoining  parts  and  the  result  is 
that  the  fine  details  of  shape  become  obliterated.  An  ordinary 
telephone  line  converts  the  pure  electric  waves  which  start  out 
from  the  sending  station  into  impure  waves,  thus  tending  to  de- 
stroy the  sharpness  of  detail  and  making  it  impossible  to  transmit 
articulate  speech  if  the  line  is  long  ;  whereas  a  telephone  Hne  that 
is  **  loaded  "  *  tends  to  keep  the  waves  pure  so  that  details  of 
shape  are  retained  as  the  wave  travels  over  the  line,  and  clear 
articulate  speech  can  be  reproduced  at  the  distant  end  of  the 
line. 

135.  Wave  pulses  in  a  canal.  —  A  consideration  of  the  simplest 


moving  WaUr    %.^^^.^,, 


W  moving  water  ^ 

still  water        \_                           M 

u:.:-  •::  i--  v:  Vi-B -^ 

slowly 
moving 
gate 

^^ 

Fig.  149.  Fig.  150. 

kind  of  wave  motion  in  a  canal,  namely,  the  kind  in  which  the 
only  perceptible  motion  of  the  water  in  the  wave  is  a  uniform 
horizontal  flow,  will  serve  better  than  anything  else  as  an  intro- 
duction to  a  discussion  of  wave  motion  along  rods,  air  columns, 
and  stretched  strings. 

The  simplest  basis  for  the  discussion  of  the  propagation  of  a 
wave  along  a  canal  is  as  follows.  Water  flows  along  a  canal  of 
rectangular  section  at  a  depth  of  x  feet  and  at  a  uniform  velocity 
(small)  of  V  feet  per  second.  A  gate  is  suddenly  closed  ;  the 
moving  water,  in  being  brought  to  rest  against  the  gate,  heaps 

*  A  **  loaded"  telephone  line  is  a  line  in  which  coils  of  wire  wound  on  iron  cores 
are  inserted  at  intervals.     This  arrangement  is  due  to  Heaviside  and  to  Pupin. 


WAVE   MOTION   AND   OSCILLATORY   MOTION.  255 

up  to  a  definite  depth  x -i-  k,  as  shown  in  Fig.  149  ;  and  a  wave 
of  arrest,'^  W,  moves  along  the  canal  at  a  definite  velocity  V. 
The  water  at  a  given  point  in  the  canal  continues  to  move  with 
unchanged  velocity  until  the  zi/ave  of  arrest  reaches  that  point, 
when  the  water  suddenly  comes  to  rest  and  heaps  up  to  the  height 
X  -\-  h.     The  velocity  of  propagation  of  the  wave  of  arrest  W  is 

F=  Vjx  (74) 

where  V  is  expressed  in  feet  per  second,  g  is  the  acceleration  of 
gravity,  and  x  is  the  depth  of  the  water  in  the  canal  in  feet.  ^  It 
is  interesting  to  note  that  the  velocity  V  of  the  wave  is  the 
velocity  that  would  be  gained  by  a  body  falling  freely  through 
the  distance   x\2. 

Proof  of  equation  (7^). —  Let  b  be  the  breadth  of  the  canal.  Consider  a  trans- 
verse slice  of  water  one  foot  thick.  The  volume  of  this  slice  is  bx  cubic  feet,  and 
its  mass  is  dbx  pounds,  where  d  is  the  density  of  the  water  in  pounds  per  cubic  foot. 
Therefore  the  kinetic  energy  of  this  slice  of  water  when  it  is  moving  at  a  velocity  of  v 
feet  per  second  is  i  jzg^dbxv'^^  according  to  equation  (27).  When  the  wave  of 
arrest,  W,  Fig.  149,  reaches  the  slice  of  water  under  consideration,  the  slice,  as  it 
comes  to  rest,  is  squeezed  together  and  increased  in  depth  to  x  -\-  h.  The  slice  is  de- 
creased in  thickness  in  proportion  to  its  increase  in  depth,  so  that  its  thickness  is  re- 
duced to  x\{x-\-  h),  or  I  —  hjx,  of  a  foot,  since  h  is  very  small.  Therefore  the 
decrease  of  thickness  is  xj h  of  a  foot.  The  force  acting  to  reduce  the  thickness  of 
the  slice  is  to  be  considered  as  that  force  which  is  due  to  the  increase  of  pressure  in 
the  water  produced  by  the  increasing  depth,  h.  This  increase  of  pressure  is  equal  to 
hd  when  the  slice  has  reached  its  greatest  height,  so  that  the  average  increase  of 
pressure  due  to  increasing  depth  is  ^td,  which  produces  over  the  face  of  the  slice  a 
force  equal  to  ^hdy^  bx,  and  the  product  of  this  force  and  the  decrease  of  thickness 
of  the  slice  gives  the  work  done  in  decreasing  its  thickness.  Therefore,  since  this 
work  is  equal  to  the  original  kinetic  energy  of  the  slice,  we  have  : 

*  The  idea  of  the  zvave  of  arrest  and  of  the  wave  of  starting  is  so  important  in  the 
discussion  of  wave  pulses  that  it  is  worth  while  to  illustrate  it  as  follows.  Imagine  a 
troop  of  soldiers  to  be  marching  along  in  single  file  at  a  distance  of  three  feet  apart, 
and  imagine  every  soldier  to  continue  to  march  as  long  as  there  is  space  in  front  of 
him.  If  the  front  man  in  the  troop  is  suddenly  stopped,  the  other  men  are  stopped 
in  succession  as  they  come  against  each  other,  and  the  stopping  occurs  at  a  point 
which  travels  uniformly  from  the  front  to  the  rear  of  the  colum,  a  wave  of  arrest,  as 
it  were.  If  the  front  man  then  starts,  the  other  men  in  the  column  start  in  succes- 
sion, and  the  starting  occurs  at  a  point  which  travels  uniformly  from  the  front  to  the 
rear  of  the  column,  a  wave  of  starting,  as  it  were. 


256-  ELEMENTS   OF   MECHANICS. 

dbxv^  -^  \dbh^ 


or 


V' 


gh^ 


X 


(i) 


Consider  the  instant  t  seconds  after  the  closing  of  the  gate  in  Fig.  149.  The  wave 
of  arrest  ^has  reached  a  distance  Vt  from  the  gate,  and  the  excess  of  water  that  is 
represented  by  the  raising  of  the  water  level  (=  VtY^hy^b  cubic  feet)  is  the 
amount  of  water  supplied  by  the  flow  of  the  canal  in  t  seconds  {^=bxvt  cubic  feet). 
Therefore 

Vthb  =  bxvt 
or 

whence,  substituting  the  value  of  v  from  equation  (i)  in  equation  (ii),  we  have  equa- 
tion (74). 

Precisely  the  same  action  that  is  represented  in  Fig.  149  may 
be  produced  in  a  still-water  canal  by  moving  the  gate  along  the 
canal  like  a  piston  at  a  low  velocity,  v^  as  indicated  in  Fig.  1 50. 


moving 
still  water ^t=^_^^---_--,^ stiJ/  ^^ter  l 


gate 
stationary 


Fig.  151. 

The  water,  in  being  set  in  motion,  heaps  Up  to  a  definite  depth 
(x  +  k),  and  a  wave  of  starting^  W,  moves  along  the  canal  at  a 
definite  velocity,  V.  If  the  gate  is  suddenly  stopped,  the  wave 
of  starting,  W,  continues  to  move  along  as  before,  the  water  next 
to  the  gate,  in  being  stopped,  drops  to  its  normal  depth,  x,  and  a 
wave  of  arrest,  W ,  moves  along  the  canal  as  shown  in  Fig.  151. 
The  uniformly  moving  and  uniformly  elevated  body  of  water, 
A,  constitutes  what  is  called  a  complete  wave  or  simply  a  wave, 
the  water  in  front  of  the  wave  is  continually  set  in  motion  at 
velocity  v  and  raised  to  the  depth  x  -f  h,  the  water  in  the  back 
part  of  the  wave  is  continually  brought  to  rest  and  lowered  to  the 


WAVE   MOTION   AND   OSCILLATORY   MOTION.  257 

normal  depth,  x^  and  thus  the  state  of  motion  which  constitutes 
the  wave  A  travels  along  the  canal  without  changing  its  character, 
friction  being  neglected. 

An  essential  feature  of  a  wave  which  moves  along  a  canal  with- 
out changing  its  shape,  is  that  the  kinetic  energy  due  to  the  uniform 
velocity  v  is  equal  to  the  potential  energy  due  to  the  elevation  h. 
When  this  relation  obtains  the  wave  is  called  a  pure  wave^ 
and  when  this  relation  does  not  obtain  the  wave  is  called  an 
impure  wave. 

The  behavior  of  an  impure  wave  pulse  in  a  canal  may  be 
understood  by  considering  an  extreme  example  of  an  impure 
wave  as  follows :  Consider  an  elevated  portion  of  still  water  in  a 
canal,  as  shown  in  Fig.  1 5  2.     This  body  of  elevated  water  is  an 

still  water  ...  -nr      „        r        XXT 

still  water  \..  •.-...  ;.  .... .,.  .■  -.j    still  water  *L ^_-^--Jy:^J^--^^-I: 

Fig.  152.  Fig.  153. 

impure  wave  inasmuch  as  its  velocity  of  flow,  v,  is  zero,  and 
therefore  the  potential  energy  of  elevation  is  of  course  not  equal 
to  the  kinetic  energy  of  flow.  Such  an  elevated  portion  of  still 
water  breaks  up  into  two  oppositely  moving  pure  waves,  and  the 
initial  stage  of  this  process  of  breaking  up  is  indicated  in  Fig.  153. 
When  a  wave  like  A,  Fig.  151,  travels  along  a  canal,  the  veloc- 
ity of  flow,  V,  is  decreased  by  friction,  whereas  there  is  no  action 
tending  to  reduce  the  elevation  h.  The  result  is  that  the  portion 
of  the  elevation  which  is  in  excess  of  what  is  required  to  consti- 
tute a  pure  wave  with  what  remains  of  the  velocity  of  flow, 
behaves  exactly  like  the  elevation  A  in  Fig.  152,  that  is,  this 
excess  of  elevation  breaks  up  into  two  pure  waves  a  and  b,  Fig. 
153,  the  portion  a  merges  with  the  original  wave  A,  and  the  por- 
tion b  shoots  backwards. 
17 


258 


ELEMENTS   OF   MECHANICS. 


Figure  1 54  represents  on  an  exaggerated  scale,  a  pure  wave, 
Ay  started  at  a  given  point  on  a  canal.  The  velocity  of  flow,  z^, 
in  this  wave  is  continually  reduced  by  friction  as  the  wave  travels 
along,  and  the  excess  of  elevation  which  is  being  thus  continually 
left  in  the  wave  causes  a  long-drawn-out  wave  to  shoot  backwards, 


A 


.  direction  of  progression 


=. S. 


bead 


tail  of  wave 

Fig.  154. 


and  after  a  time  the  wave  takes  on  the  form  shown  by  BB. 
The  tail  of  the  wave  extends  far  back  of  the  original  starting 
point  of  the  wave. 

If  a  canal  is  brimful  of  water  so  that  the  elevation  causes  an 
overflow  or  spill,  the  tendency  is  for  a  wave  to  remain  pure  and 
therefore  to  be  propagated  without  change  of  shape,  because  the 


Fig.  155. 

elevation  is  reduced  by  spill  and  the  velocity  of  flow  v  is  reduced 
by  friction.  This  is  precisely  analogous  to  the  action  which  takes 
place  on  a  poorly  insulated  telephone  line  and  causes  such  a  tele- 
phone line  to  transmit  speech  more  distinctly  than  the  same  line 
would  if  it  were  thoroughly  insulated. 


WAVE   MOTION   AND   OSCILLATORY   MOTION.  259 

136.  Wave  pulses  along  a  steel  rod. — Before  proceeding  to  dis- 
cuss the  production  of  a  complete  wave  of  compression  (or  stretch) 
along  a  steel  rod,  it  is  of  interest  to  consider  the  behavior  of  a 
steel  rod  when  it  is  moving  endwise  at  a  small  velocity,  7/,  and 
strikes  a  rigid  wall,  as  shown  in  Fig.  155.  The  portion  of  the 
rod  next  to  the  wall  is  compressed  and  immediately  brought  to 
rest,  and  a  wave  of  arrest,  W,  moves  along  the  rod,  as  indicated. 

A  _  ™°^jg W    sun B_  J 

^^■^^g:^^^^g.-'.-  'i  •'.'.  •.;.  •.*•'.  i^ '/compressed'.'-'.'/^  '•:■*•'"."*.*:*  •'.'.•'.''  ii"'^^ 


W 


Fig.  156. 


At  the  instant  that  the  wave  of  arrest  reaches  the  free  end,  A, 
of  the  rod  in  Fig.  155,  the  entire  rod  is  stationary  and  uniformly 
compressed.  Therefore  balanced  forces  act  on  every  portion  of 
the  rod  except  the  layer  of  material  at  the  extreme  end,  A,  so 
that  this  layer  immediately  starts  to  move  at  reversed  velocity,  v 
(away  from  the  wall),  as  indicated  in  Fig.  1 56,  and  a  wave  of 
starting  travels  back  along  the  rod,  as  indicated  in  the  figure. 


still 


^ 

wall 


!•: .'  •.*•  .stretched  '•; : 


w 

Fig.  157. 

When  this  wave  of  starting  reaches  the  end  B  of  the  rod,  the 
entire  rod  is  free  from  compression  and  is  moving  away  from  the 
wall  at  velocity,  v. 

If  the  rod  be  glued  fast,  as  it  were,  to  the  wall,  the  end  B  of 
the  rod  is  immediately  brought  to  rest  and  stretched,  and  a  ivave 
of  arrest  travels  along  the  rod  as  shown  in  Fig.  157.  When 
this  wave  of  arrest  reaches  the  free  end,  A,  of  the  rod,  the  entire 


26o 


ELEMENTS   OF   MECHANICS. 


rod  is  stationary  and  uniformly  stretched,  so  that  the  end  layer 
at  A  immediately  begins  to  move  towards  the  wall,  and  a  wave 
of  starting  travels  back  towards  the  wall  as  indicated  in  Fig.  158. 
When  this  wave  of  starting  reaches  B^  the  entire  rod  is  precisely 
in  its  initial  condition,  namely,  moving  towards  the  wall  at  uni- 
form velocity  v.     The  action  here  described  takes  place  so  rapidly 


A      moving           J^ 

still 

B 

L-i:::=-=^^-^-^:=H 

i 

. :  ,*•• .;  \  stretched  •-".';;.  .v..'.*: ;  > 

-■•.•':  :*.:•'•■••.•.•: 

iw^sis-i 

wall 


w 


Fig.  158. 


that  it  cannot  be  followed  with  the  eye.  In  fact,  the  wave  of 
arrest  diVid  the  wave  of  starting  m  Figs.  155  to  158  travel  at  a 
velocity  of  16,950  feet  per  second,  so  that  the  entire  cycle  of 
movements  above  described  would  take  place  424  times  per  sec- 
ond in  a  steel  rod  10  feet  long,  inasmuch  as  the  waves  of  arrest 


A  moving 


wall 


not  stretpbed 


and  starting  must  travel  over  the  rod  four  times  in  one  complete 
cycle  of  movements. 

A  helical  spring  may  be  made  to  perform  the  same  series  of 
movements  as  the  steel  rod  as  above  described,  and  at  a  much 
slower  rate.  Thus,  if  one  end  of  a  helical  spring  be  attached  to 
a  wall,  as  shown  in  Fig.  1 59  ;  the  spring  may  be  uniformly 
stretched  by  pullmg  on  the  end  A.  If  the  spring  is  then  released 
the  free  end  begins  to  move  towards  the  wall,  and  a  zvave  of  start- 
ing travels  towards  the  wall  exactly  as  described  in  connection 


WAVE   MOTION   AND   OSCILLATORY   MOTION.  261 

with  Fig.  158  ;  then  a  zvave  of  arrest  travels  back  from  B  to  A 
exactly  as  described  in  connection  with  Fig.  1 5  5  ;  a  wave  of 
starting  (  velocity  v  away  from  the  wall)  then  travels  from  A  to 
B  ;  and  a  wave  of  arrest  then  travels  from  B  to  A,  bringing  the 
spring  into  its  initial  uniformly  stretched  condition. 

This  example  of  the  traveling  of  waves  of  starting  and  arrest 
along  a  steel  rod  or  a  helical  spring  illustrates  one  of  the  most 
important  applications  of  the  ideas  of  wave  motion,  namely, 
their  application  to  the  study  of  oscillatory  motion.  In  the  dis- 
cussion of  the  oscillatory  motion  of  a  weight  fixed  to  the  end  of 
a  spring  (Art.  43),  and  in  the  discussion  of  the  oscillatory  motion 
of  a  torsion  pendulum  (Art.  6^^,  the  elastic  part  of  the  system 
is  assumed  to  have  negligible  mass,  and  the  massive  part  of  the 
system  is  assumed  to  have  negligible  elasticity,  or,  in  other  words, 
the  mass  is  assumed  to  be  concentrated  in  one  part  of  the 
oscillating  system  and  the  elasticity  is  assumed  to  be  concentrated 
in  another  part  of  the  oscillating  system.  The  oscillatory  motion 
of  such  a  system  is  extremely  simple  in  comparison  with  the 
infinite  variety  of  oscillatory  movements  that  may  be  performed 
by  a  given  *  system  in  which  elasticity  and  mass  are  distributed 
throughout  the  system,  f 

The  velocity  of  propagation  of  the  waves  of  arrest  and  starting  in  a  steel  rod  may 
be  derived  as  follows  :  After  a  certain  interval  of  time  /  the  wave  of  arrest  IV, 
reaches  the  end  A  in  Fig.  155,  having  traveled  over  the  length  of  the  rod  Z.  During 
this  time,  /,  the  end  B  of  the  rod  has  been  stationary  and  the  end  A  has  been  mov- 
ing steadily  at  velocity  v.  Therefore  the  rod  has  been  shortened  by  the  amount  / 
^=vt,  so  that  the  shortening  per  unit  length  Ij L\^^=(i  of  equation  (50)]  is  equal  to 
vtj  L.  The  kinetic  energy  in  foot-pounds  of  the  moving  rod  before  it  strikes  the  wall 
is  i/ag'X  Lad^  ^'2,  according  to  equation  (27),  where  d  is  the  density  of  the  rod 
in  pounds  per  cubic  foot  and  a  is  its  sectional  area  in  square  feet.  At  the  instant  that 
the  wave  of  arrest  reaches  the  end  A,  the  potential  energy  of  the  compressed  rod  is 
y^E^'^y^  La,  according  to  equation  (50),  where  E  is  the  stretch  modulus  of  the 

■'*•  See  Art.  142. 

f  Th^  mode  of  treatment  of  an  oscillating  system  with  concentrated  mass  and  con- 
centrated elasticity  is  exemplified  by  the  simple  theory  of  alternating  currents  (con- 
centrated inductance  and  capacity),  and  the  mode  of  treatment  of  an  oscillating 
system  with  distributed  mass  and  distributed  elasticity  in  terms  of  wave  motion  is  ex- 
emplified by  the  theory  of  transmission  lines  for  alternating  currents  (distributed  in- 
ductance and  distributed  capacity.). 


262 


ELEMENTS   OF   MECHANICS. 


steel  in  pounds  per  square  foot  (=  144  X  30>oo*^>°°°)»  ^"^  ^^  ^^  ^^^  volume  of 
the  rod.  This  potential  energy  of  compression  is  equal  to  the  original  kinetic  energy 
of  the  rod  so  that 


so  that 


^^^Ladv'^^lEi^^  La  =  \E 


La 


/2   —  ~    d 


(75) 
where  Fis  required  velocity  of  progression  of  the  wave  of  arrest  in  Fig.  155. 

A  completely  self-contained  wave  of  compression  may  be  pro- 
duced in  a  steel  rod  in  a  manner  exactly  analogous  to  the  pro- 
duction of  a  complete  water  wave  in  a  canal  by  moving  a  gate  as 
indicated  in  Fig.  150.  One  end  of  a  long  steel  rod  AB,  Fig. 
160  is  struck  with  a  hammer,  and  let  us  assume,  for  the  sake  of 


IT 

jr 


axis  of  progression 


A     still 


W 


moving 


w 


still 


^V-^'iCi':--'.'-::::]^!;;^ 


compressed 


jn-r  not   compressed 
Fig.  160. 


simplicity,  that  the  hammer  continues  to  move  at  a  constant 
velocity  v  while  it  pushes  on  the  end  of  the  rod,  and  then  re- 
bounds. The  result  of  such  a  hammer  blow  would  be  to  set  up 
a  wave  of  starting  W,  the  portion  of  the  rod  behind  IV  would 
be  moving  at  uniform  velocity  z>  and  would  be  uniformly  com- 
pressed. At  the  instant  of  rebounding  of  the  hammer,  however, 
the  compression  in  the  moving  portion  of  the  rod  would  cause  the 
end  layer  B  of  the  rod  to  stop,  and  a  wave  of  arrest  W  would 
follow  the  wave  of  starting  W^as  indicated  in  the  figure. 

The  above  description  applies  to  the  production  of  a  wave  of 
compression  in  a  steel  rod.     A  sudden  pull  on  the  end  B  of  the 


WAVE   MOTION   AND   OSCILLATORY   MOTION. 


263 


rod  in  Fig.  160  would  produce  a  completely  self-contained  wave 
of  extension  or  stretch.     Such  a  wave  is  shown  in  Fig.  161. 

The  complete  waves  A' A'  in  Figs.  1 60  and  161  are  represented 
graphically  by  the  small  rectangles,   aa^   the  ordinates  of  which 


axis  of  progression 

a 

Still         B 

A    still      1 

r 

moving 

: 
: 

w' 

'.  •' .' '.'  ',•'■'..'•'•' '  .'•' ' 

sudden  pull 
{no  longer  acting) 

not         \ 
Stretched. 

V 

stretched 

Fig.  161. 

not  stretched 

represent  the  uniform  velocity  of  motion  v  of  the  portion  of  the 
rod  which  is  in  the  wave. 

Reflection  with  or  without  phase  reversal.  —  An  action  which  is 
very  important  in  the  theory  of  reflection  of  light  and  sound,  is 


c 

-^^  compression  -^ 

Compression  everywhere  zerolj                   C 

axis 

of  prog 

ression 

axis  of  progression 

t- 

^ 

1 

f 

M 

* 

M 

axis 

of  prog 

ression 

axis  of  progression 

the  action  which  is  called  reflection  with  or  without  phase  reversal, 
as  the  case  may  be.  This  action  may  be  easily  understood  by 
considering  the  reflection  of  a  complete  wave  like  A '  of  Fig.  1 60 


264 


ELEMENTS    OF   MECHANICS. 


or  Fig.  161.  Let  us  consider  first  the  reflection  of  a  complete 
wave  from  the  free  end  of  a  rod  as  shown  in  Figs.  162  to  165. 
At  the  moment  when  the  front  of  the  complete  wave  reaches  the 
end  of  the  rod,  the  compression,  being  suddenly  relieved,  is  con- 
verted into  motion  of  the  end  layer,  which  motion,  added  to  the 
previously  existing  motion  in  the  wave,  causes  a  doubled  velocity 
of  the  end  layer.  This  doubled  velocity  establishes  itself  by  a 
wave  of  starting,  WW,  as  shown  in  Fig.  162,  and  the  region 
in  which  this  doubled  velocity  exists  is  entirely  freed  from  com- 
pression. A  moment  later,  when  the  wave  of  starting  W  W^ 
and  the  wave  of  arrest    WW  in  Fig.    162  become  coincident, 


axis  of  progression 


axis  of  progression 


Fig.  164. 


Fig.  165. 


the  entire  energy  of  the  original  wave  is  represented  by  the 
doubled  velocity  of  a  portion  of  the  rod  one-half  as  long  as  the 
original  complete  wave,  and  the  compression  is  everywhere  zero, 
as  shown  in  Fig.  163.  The  motion  of  the  end  of  the  rod  as 
shown  in  Fig.  163  begins  to  produce  a  stretch  in  the  region  near 
W'W",  causing  the  doubled  velocity  on  one  side  of  W'W^ 
to  be  reduced  to  the  normal  value  v,  and  setting  the  material  to 
the  right  of  W'W'  in  motion.  A  wave  of  semi-arrest,  WW, 
Fig.  164,  travels  towards  the  end  of  the  rod  and  the  wave  of 


WAVE   MOTION   AND    OSCILLATORY   MOTION. 


265 


Starting^  W'W\  travels,  as  indicated  in  Fig.  164.  When  the 
wave  of  semi -arrest  WW^  Fig.  1 64,  reaches  the  end  of  the  rod, 
the  combined  action  of  the  uniform  stretch  and  the  uniform 
velocity  v  of  the  portion  of  the  rod  which  constitutes  the  reflected 
(complete)  wave  brings  the  rod  to  rest  at  the  extreme  end  and 
frees  it  from  distortion  as  the  reflected  (complete)  wave  moves 
towards  the  right  in  Fig.  165.  The  diagrams  C  and  M  are 
graphical  representations  of  the  compression  (or  stretch)  and  of 
the  motion  v  of  the  material  of  the  rod. 

The  details  of  reflection  of  a  complete  wave  of  stretch  or  ex- 
tension from  the  free  end  of  a  rod  are  very  similar  to  the  details 
of  reflection  of  a   complete  wave  of  compression  as   described 


l^cowpression 


axis  of  progression 


f 

compression 

I 


axis  of  progression 


M 


axis  of  progression 


!  I 

I  ; 

!  (tnotion  everyMrbere  zero^  J^ 

1 '  axis  of  progression 

i 


''kid 


rigid 
wall 


^ 


'•'-.'vi". 


Fig.  166. 


Fig.  167. 


above.  The  reflection  of  either  kind  of  a  wave  from  the  free  end 
of  a  steel  rod  results  in  the  conversion  of  compression  into  stretch 
or  the  conversion  of  stretch  into  compression,  leaving  the  velocity, 
V,  in  the  wave  unchanged  in  direction.  Therefore  this  kind  of 
reflection  is  called  reflection  without  reversal  of  velocity-phase 
(but  with  reversal  of  distortional-phase). 

The  details  of  reflection  of  a  complete  wave  of  compression  from 
the  end  of  a  steel  rod  which  rests  against  a  rigid  wall  are  shown 
in  Figs.  166  to  169.     When  the  original  complete  wave  reaches 


266 


ELEMENTS   OF   MECHANICS. 


the  end  of  the  rod  the  motion  in  the  wave  compresses  the  end 
layer  of  the  rod,  which  compression  is  added  to  the  previously- 
existing  compression  in  the  wave,  giving  a  double  compression  in 
the  end  layer  of  the  rod,  the  velocity  in  this  layer  being  reduced 
to  zero.     The  details  of  the  entire  action  may  be  seen  from  the 


1 ~ 

\^  compression 

A 


axis  of  progression 
Jf  axis  of  progression 


recompression 

i 


axis  of  progression 
axis  of  progression    M 


Fig.  168. 


Fig.   169. 


figures.  The  details  of  reflection  of  a  complete  wave  of  stretch 
or  extension  from  the  rigid  end  of  a  rod  are  not  very  different 
from  the  details  of  reflection  of  a  complete  wave  of  compression. 
The  reflection  of  either  kind  of  a  wave  from  the  rigid  end  of  a  rod 
results  in  the  reversal  of  the  velocity  v,  without  converting  com- 


compressed 

Fig.  170. 

pression  into  stretch  or  stretch  into  compression.  Therefore  the 
reflection  from  the  rigid  end  of  a  rod  is  called  reflection  with  re- 
versal of  velocity -phase  (but  without  reversal  of  distortional-phase). 
137.  Wave  pulses  in  air  and  water  pipes.  Figure  170  repre- 
resents  a  long  pipe  filled  with  air.     If  the  piston  is  moved  at  a  small 


WAVE   MOTION    AND   OSCILLARORY   MOTION.  267 

velocity  v  for  a  short  time  and  then  brought  to  rest,  a  complete 
wave  of  compression  is  produced  which  travels  along  the  tube,  as 
shown  in  the  figure.  This  wave  of  compression  is  similar  in 
every  detail  to  the  complete  wave  of  compression  which  is  pro- 
duced in  a  steel  rod  by  striking  the  end  of  the  rod  with  a  ham- 
mer. If  the  piston  P  is  moved  at  a  small  velocity  v  in  an  out- 
ward direction  and  then  stopped,  a  complete  wave  of  rarefaction 
is  produced. 

An  interesting  phenomenon  of  wave  motion  occurs  when  a 
wide  open  hydrant  is  suddenly  closed.  The  moving  water  is 
suddenly  brought  to  rest  against  the  valve  and  raised  to  a  very 
high  pressure,  and  a  T^^^^T^^  <?/"  «rr^.$-/,    WW^   Fig.  1 7 1 ,  travels  along 


Miiiitt 


fiot  comfre^sed  J^      compressed 


Fig-.  171. 


the  supply  pipe,  as  shown.  At  the  point  where  the  small  supply 
pipe  widens  out  into  a  large  street  main,  the  action  is  very  similar 
to  the  action  which  takes  place  at  the  free  end  of  a  steel  rod,  as 
described  in  Art.  136.  Therefore  when  the  wave  of  arrest, 
WW,  Fig.  171,  reaches  the  street  main,  the  uniformly  com- 
pressed water  in  the  supply  pipe  starts  the  water  moving  back- 
wards into  the  main,  this  motion  being  established  by  a  wave  of 
starting  which  travels  from  the  street  main  to  the  hydrant.  When 
this  wave  of  starting  reaches  the  hydrant,  the  water  in  the  supply 
pipe  is  moving  towards  the  street  main  at  a  uniform  velocity. 
This  backward  movement  of  the  water  is  stopped  immediately  in 
the  neighborhood  of  the  valve,  producing  a  great  decrease  of 
pressure  there,  and  a  wave  of  arrest  again  travels  from  this  valve 
to  the  water  main.  When  this  wave  of  arrest  reaches  the  water 
main,  the  water  in  the  supply  pipe  is  uniformly  expanded,  and  a 
wave  of  starting  then  travels  from  the  street  main  to  the  valve. 


268 


ELEMENTS   OF   MECHANICS. 


setting  the  water  in  motion  towards  the  valve,  as  at  first.  This 
motion  of  the  water  is  suddenly  stopped  by  the  valve,  causing  a 
sudden  rise  of  pressure,  as  at  first.  This  action  is  often  repeated 
five  or  six  times  when  a  hydrant  is  closed,  producing  five  or  six 
sharp  clicks  in  succession  as  the  water  is  repeatedly  brought  to 
rest  against  the  closed  valve  of  the  hydrant. 

138.  Wave  pulses  along  wires  and  strings.  —  Imagine  a  long 
stretched  wire,    AB,    Fig.  172,  to  be  moving  sidewise  at  a  small 


_-L_i:.L-i-j-lJ— I 


r      i      >^      f      f      Y 


4—l-s 


Fig.  172. 


velocity,  v,  and  to  strike  a  rigid  support  or  bridge,  »S.  The  por- 
tion of  the  wire  immediately  contiguous  to  5  is  stopped  at  once 
and  two  waves  of  arrest,  WW,  travel  along  the  wire  at  a  definite 
velocity  V,  as  indicated  in  the  figure.  The  portion  of  the  wire  to 
the  left  of  W\Vi  Fig.  173  continues  to  move  sidewise  as  if  nothing 


Fig.  173. 

had  happened,  the  portion  of  the  wire  to  the  right  of  W  in  Fig. 
173  is  left  in  an  inclined  direction,  and  the  portion  OS  of  the 
moving  wire  is  stretched  to  the  length  SW  m.  being  brought  to 
rest. 

A  complete  wave  pulse  may  be  produced  on  a  wire  as  follows  : 
Imagine  a  wire  to  be  stationary  in  the  position  A' B' ,  Fig.  172, 
and  imagine  a  hammer  5  to  come  upwards  at  velocity  v,  touch 
the  wire  and  continue  to  move  at  velocity  v  for  a  short  time,  and 
then  rebound,  leaving  the  wire  free.  The  result  of  the  first  im- 
pact of  5  against  the  wire  is  to  set  up  two  waves  of  startingy 


\ 


WAVE   MOTION   AND   OSCILLATORY   MOTION.  269 

WW  (these  are  waves  of  arrest  when  we  imagine  the  wire  to 
move  and  5  to  be  stationary).  Then  when  the  hammer  5  re- 
bounds, the  tension  of  the  incHned  portions  of  the  wire  brings  the 
wire  at  5  to  rest  at*once,  and  two  waves  of  arrest,  W'  W\  travel 
outwards  from  5,  as  indicated  in  Fig.  174. 

It  is  especially  interesting  to  consider  the  details  of  motion  of 
a  guitar  string  which  is  pulled  to  one  side,  by  placing  tiie  finger 


at  the  middle  of  the  string,  and  then  released.  The  form  of  the 
string  at  the  instant  of  release  is  shown  by  A^B^,  Fig.  175.  The 
tension  of  the  string  sets  the  point  /  in  motion  at  the  instant  of 
release,  and  two  waves  of  starting,  WW,  travel  to  the  ends  of 
the  string  as  shown  by  the  diagram,  A^B^.  When  these  waves 
of  starting  reach  the  ends  of  the  string,  the  whole  wire  is  moving 
sidewise  at  uniform  velocity  v,  and  waves  of  arrest,  W'  W\  travel 
from  the  ends  of  the  string  towards  the  middle  as  shown  in  the 
sketch  A^B^.  When  these  uaves  of  arrest  reach  the  middle  of 
the  string  the  entire  string  is  momentarily  stationary  in  the  con- 
figuration shown  by  the  sketch,  AJ3^.  Then  the  previous  action 
is  repeated  in  reverse  order.  If  the  string  is  stiff,  if  the  end  sup- 
ports A  and  B  are  not  rigid  (of  course  they  are  not  in  a  guitar), 
and  if  the  air  friction  is  considerable,  this  extremely  simple  oscil- 
latory motion  of  the  string  is  greatly  modified  as  the  string  con- 
tinues to  vibrate. 

The  motion  of  a  guitar  string  which  is  plucked  near  one  end 
is»represented  in  Fig.  176.  The  uniformly  moving  straight  por- 
tion of  the  string  is  equally  inclined  to  the  two  stationary  parts 
of  the  string. 

139.  Maximum  strain  in  suddenly  loaded  structures.  When 
a    load    is    slowly  applied  to    a  structure,   the  strain  increases 


2/0 


ELEMENTS   OF   MECHANICS. 


slowly  until  the  structure  is  in  equilibrium  under  the  load.    This 
value  of  the  strain  is  called  the  eqnilibriuin  strain^  or  the  static 


a,k\  \  \\  \  \\  \  \  \  \  \  \  \  \  \  w  \  \  \  \  \^, 


<t  1 1 1  l.jj 

44.t    f   t   t  f   t    t    f    t   t    t  t   t    t    t  t    t   f    t    f    t 


^-^^ 


nrr 


Fig.  175. 


strain^  corresponding  to  the  load.     When  a  load  is  suddenly  ap- 
plied to  a  structure,  the  momentum  which  the  structure  gains 


WAVE   MOTION   AND   OSCILLATORY   MOTION. 


27 


while  the  strain  is  increasing  to  the  equilibrium  value  carries  the 
strain  beyond  the  equilibrium  value.      The  maximum  strain  pro- 


Fig.  176. 

duced  by  a  suddenly  applied  load  is  equal  to  two  times  the  equilib- 
rium strain  corresponding  to  the  load,  friction  being  negligible. 

Case  I.  When  the  mass  of 
the  elastic  structure  is  negligible 
in  comparison  with  the  mass  of 
the  load.  Figure  177  represents 
a  helical  spring  in  its  unstretched 
initial  position,  and  a  weight 
attached  to  the  spring,  but  sup- 
ported by  the  hand.  When  the 
weight  is  released,  it  oscillates 
up  and  down,  coming  back  to 
its  initial  position,  or  nearly  to 
its  initial  position,  repeatedly. 
But  the  equilibrium  position  of 
an  oscillating  body  is  midway  be- 
tween its  extreme  positions,  and 
therefore  the  lowest  position 
reached  by  the  oscillating  weight 
in  Fig.  1 77  is  at  a  distance,  2x, 
below  the  initial  position,  where 
X  is  the  distance  between  the  equilibrium  position  and  the  initial 
position,  as  shown. 


'f^imtial  portion' 


J*. Jequillbzlam  position 

L J         a; 


'k-. 


.j[  extreme  position 


L I 


Fig.  177. 


2/2 


ELEMENTS   OF   MECHANICS. 


Case  2.  When  the  mass  of  the  elastic  structure  is  not  negli- 
gible. *  A  force,  F^  is  suddenly  applied  to  the  top  of  a  steel  column, 
as  shown  in^Figs.  178  and  179.  The  top  of  the  column  is  im- 
mediately compressed  by  the  full  value  of  the  force  (equilibrium 
strain),  and  also  set  moving  downwards  at  a  definite  velocity,  v. 
This  equilibrium  compression  and  the  downward  velocity,  v, 
are  imparted  to  the  successive  portions  of  the  column  by  the 
wave  of  starting^  W,  which  travels  downwards,  as  indicated  in 


moving  and  compresseit 


still  and 

not  compressed 


Fig.  178. 


moving  and  compressed 


still  and 
doubly 


Fig.  179. 


Fig.  178.  At  the  instant  when  this  wave  reaches  the  bottom  of 
the  column,  the  column  is  everywhere  compressed  to  the  equilib- 
rium value ;  but  its  downward  motion  creates  a  double  compres- 
sion at  the  bottom,  and  this  double  compression  extends  upwards 
with  the  upward  movement  of  the  wave  of  arrest^  W,  Fig.  179. 
When  this  wave  of  arrest  reaches  the  top  of  the  column,  the  en- 
tire column  is  under  a  strain  twice  as  great  as  the  equilibrium 

*  In  the  discussion  of  the  case  here  given,  the  load  is  assumed  to  have  a  neghgible 
mass.  Substantially  the  same  conclusion  would  be  reached,  if  the  mass  of  the  load 
were  taken  into  account,  but  the  argument  would  be  very  complicated. 


WAVE   MOTION   AND   OSCILLATORY   MOTION.  273 

strain  corresponding  to  the  load.     The  subsequent  oscillations 
of  the  column  need  not  be  considered. 

140.  Longitudinal  waves  and  transverse  waves.  —  T^e  kind  of 
water  waves  described  in  Art.   135,  and  the  waves  described  in 

I  Arts.    136  and    137  are   called  longitudinal  waves  because   the 
|i     actual  motion  of  the  water  or  steel  or  air  in  the  wave  is  parallel 

II  to  the  direction  of  progression  of  the  wave,  v  parallel  to  V. 
Sound  waves  in  the  air  are  longitudinal  waves.  The  kind  of 
waves  described  in  Art.  138  are  called  transverse  waves  because 

\^  the  actual  motion  of  the  wire  in  the  wave  (between  W  and  W  in 
Fig.  174)  is  at  right  angles  to  the  direction  of  progression  of  the 
wave,  V  at  right  angles  to  V.  Light  waves  in  the  ether  are 
transverse  waves.  In  ordinary  water  waves,  each  particle  of 
water  describes  a  circular  or  elliptical  orbit,  the  plane  of  which  is 
vertical  and  parallel  to  the  direction  of  progression  of  the  waves, 
that  is  to  say,  ordinary  water  waves  are  neither  purely  longitudinal 
nor  purely  transverse  waves. 

Polarization  of  transverse  waves.  —  Imagine  the  end  of  a  long 
stretched  rope  to  be  moved  rapidly  up  and  down  so  as  to  pro- 
duce waves  traveling  along  the  rope.  The  direction  of  oscillation 
of  the  rope  is  everywhere  in  a  vertical  plane ;  transverse  waves 
in  which  the  direction  of  oscillation  is  parallel  to  a  certain  plane 
are  called  plane  polarized  waves. 

141.  The  wave  front.  —  The  idea  of  the  wave  front  plays  a 
very  important  part  in  the  wave  theory  of  light.  Imagine  a  dis- 
turbance to  take  place  at  a  point  on  the  surface  of  a  pond.  If 
this  disturbance  is  simple  in  character,  a  sharply  defined  wave 
passes  out  from  it.  If  the  disturbance  is  complicated,  for  ex- 
ample, if  a  handful  of  pebbles  is  thrown  into  the  water,  the  waves 
in  the  immediate  neighborhood  of  the  disturbance  are  utterly  con- 
fused. At  a  great  distance  from  a  disturbance,  however,  the 
waves  always  become  sharply  defined,  whether  the  disturbance 
be  simple  or  complicated,  that  is  to  say,  the  waves  form  definite 
ridges  on  the  surface  of  the  water,  and  a  line  can  be  drawn  along 
the  water's  surface  so  as  to  pass  through  points  where  the  water 


274  ELEMENTS   OF   MECHANICS. 

surface  is  in  a  similar  state  of  motion  and  a  similar  condition  of 
distortion.  Such  a  line  is  called  a  wave  front.  In  the  case  of 
sound  and  light  waves,  the  wave  front  is  a  surface  imagined  to  be 
drawn  through  the  air  or  ether,  the  motion  and  distortion  of  the 
medium  being  the  same  at  every  point  of  this  surface.  A  wave 
which  has  a  plane  front  is  called  a  plane  wave  and  a  wave  which 
has  a  spherical  front  is  called  a  spherical  wave.  The  importance 
of  the  idea  of  wave  front  in  optics  is  due  to  the  fact  that  the 
direction  of  progression  of  the  wave  is  at  right  angles  to  the  wave 
front  when  the  medium  is  isotropic. 

142.  Free  oscillations  of  elastic  systems.  —  The  discussion  given 
in  Arts.  136  and  138  illustrate  the  application  of  the  wave  theory 
to  the  study  of  oscillatory  motion.  A  complete  development  of 
this  aspect  of  the  wave  theory  depends  upon  a  consideration  of 
wave  trains,  and  this  development  is  usually  included  in  treatises 
on  acoustics. 

Simple  modes  of  oscillation.  —  An  elastic  system,  such  as  a 
stretched  string,  an  air  column,  a  steel  rod,  or  a  bell,  may  oscil- 
late in  such  a  way  that  every  particle  of  the  system  performs 
simple  harmonic  motion  of  the  same  frequency.  When  a  system 
vibrates  in  this  way  there  are,  in  general,  certain  places  called 
nodes  where  there  is  no  motion  at  all,  and  the  vibrating  regions 
between  these  nodes  are  called  vibrating  segments.  A  system 
vibrating  in  this  way  is  said  to  vibrate  in  a  simple  mode.  A  given 
elastic  system  may  perform  a  great  variety  of  simple  modes  of 
oscillation.  For  example,  a  string  may  vibrate  as  a  whole,  or 
in  two  parts,  or  in  three  parts,  or  in  any  number  of  parts.  A 
column  of  air  in  a  tube  may  vibrate  as  a  whole,  or  in  halves,  or 
in  any  number  of  parts  ;  the  notes  of  a  bugle,  for  example,  are 
produced  by  causing  the  air  column  in  the  instrument  to  perform 
different  simple  modes  of  oscillation.  A  flat  metal  plate  clamped 
at  its  center  may  be  set  vibrating  by  drawing  a  violin  bow  across 
its  edge.  If  sand  be  strewn  upon  the  plate,  the  sand  is  thrown 
away  from  the  vibrating  segments  and  heaped  up  on  the  nodal 
lines,  forming  very  beautiful  figures,  which  were  first  studied  by 


WAVE   MOTION   AND    OSCILLATORY   MOTION. 


275 


Chladni.  Figure  180  shows  some  of  the  sand  figures  obtained 
by  Chladni  and  depicted  in  his  treatise  on  Acoustics  (Leipzig, 
1787). 

In  a  vibrating  string,  nodal  points  separate  the  vibrating  seg- 
ments ;  in  a  vibrating  plate,  nodal  lines  separate  the  vibrating 
segments ;  and  in  a  system  of  three  dimensions,  nodal  surfaces 
separate  the  vibrating  segments. 

Compound  modes  of  oscillation.  —  An  elastic  system  may  per- 
form any  number  of  its  simple  modes  of  oscillation  simultane- 
ously. Li  this  case,  the 
system  is  said  to  oscillate 
in  a  compound  mode  and 
each  particle  of  the  system 
performs  simultaneously 
the  various  simple  har- 
monic movements  corre- 
sponding to  the  respective 
simple  modes  of  oscillation 
that  are  present  in  the  oscil- 
latory motion  of  the  system. 
For  example,  a  metal  plate, 
struck  by  a  hammer,  per- 
forms simultaneously  a 
great  number  of  the  simple 
modes  of  oscillation  of 
which  it  is  capable,  and 
usually,  the    musical    tone 

corresponding  to  each  simple  mode  of  oscillation  can  be  sepa- 
rately heard.  The  simple  type  of  oscillation  of  a  steel  rod,  de- 
scribed in  Art.  136,  and  the  simple  type  of  oscillation  of  a  string 
described  in  Art.  138,  constitute  compound  modes  of  oscillation 
in  the  sense  in  which  this  term  is  here  used,  and,  as  a  matter  of 
fact,  it  is  possible  to  detect  a  series  of  distinct  tones  in  the  sound 
that  is  emitted  by  a  steel  rod  or  string  vibrating  in  the  manner 
described  in  Arts.  136  and  138. 


Fig.  180. 


2/6  ELEMENTS   OF   MECHANICS. 

143.  Forced  oscillations  of  elastic  systems.  Resonance. — When 
an  elastic  system  like  a  string  or  a  steel  rod,  is  distorted  and  re- 
leased, it  performs /r^^  oscillatio7is  at  a  definite  frequency.  When 
an  elastic  system  is  acted  upon  by  a  periodic  force  from  outside, 
the  system  is  made  to  oscillate  at  the  same  frequency  as  the  ap- 
plied force.     Such  oscillations  are  z^XX^^  forced  oscillations. 

Consider  an  oscillating  elastic  system.  If  the  oscillations  are 
free,  then  the  forces  required  to  overcome  the  inertia  of  the  mov- 
ing parts  of  the  system  as  they  are  repeatedly  stopped  and  started, 
will  be  supplied  wholly  by  the  elastic  reactions  in  the  system, 
and  the  forces  required  to  produce  the  repeated  distortion  of  the 
system  will  be  supplied  by  the  inertia  reactions  in  the  system,  so 
that  the  only  outside  force  that  would  be  required  to  maintain 
free  oscillations  of  a  system  is  a  force  sufficient  to  overcome  fric- 
tion, and  this  force  must  be  of  the  same  frequency  as  the  free 
oscillations  of  the  system.  If,  however,  the  outside  force  is  not 
of  the  same  frequency  as  the  free  oscillations  of  the  system,  a 
large  part  of  the  force  must  be  used  to  overcome  the  inertia  reac- 
tions or  the  elastic  reactions  in  the  system,  and  only  a  small  por- 
tion of  the  force  is  available  for  overcoming  friction.  Therefore  a 
periodic  force  acting  on  an  elastic  system  produces  a  maximum  of 
violence  of  oscillation  if  the  frequency  of  the  force  is  the  same  as  the 
frequency  of  the  free  oscillations  of  the  system.  This  phenomenon 
is  called  resonance.  It  is  exemplified  by  the  very  decided  oscil- 
latory motion  of  a  bridge,  which  is  produced  by  a  comparatively 
weak  force  of  the  proper  frequency.  Thus,  the  rhythmic  motion 
of  a  trotting  horse  sets  a  bridge  into  violent  oscillatory  motion  if 
the  rhythm  of  the  movements  of  the  horse  is  the  same  as  the 
rhythm  of  free  oscillation  of  the  bridge.  It  usually  requires  a 
long  continued  action  of  a  periodic  force  to  fully  establish  the 
comparatively  violent  oscillations  of  a  system  by  resonance. 

Problems. 
168.  What  is  the  most  suitable  velocity  of  propulsion  of  a  canal 
boat  in  a  canal  /  feet  deep  ? 


WAVE   MOTION   AND   OSCILLATORY   MOTION.  277 

Note. — When  a  canal  boat  is  propelled  at  the  velocity  of  wave  propagation  in  a 
canal,  it  rides  on  the  wave  which  it  produces,  and  less  force  is  required  to  propel  it 
than  would  be  required  if  the  boat  moved  at  a  considerably  less  velocity  ;  also,  the 
agitation  of  the  water  is  less  pronounced  than  it  would  be  at  considerably  less  velocity, 
and  the  washing  of  the  canal  banks  is  correspondingly  less. 

169.  A  helical  spring  3  feet  long  weighs  one-half  pound  and 
it  is  elongated  one  inch  by  the  force  of  2  pounds-weight.  One 
end  of  the  spring  is  attached  to  a  support  and  the  spring  is 
stretched  and  suddenly  released.  How  many  complete  oscilla- 
tions does  it  make  per  second  ? 

170.  Plot  a  curve  of  which  abscissas  represent  elapsed  time 
and  of  which  the  ordinates  represent  the  varying  distance  of  the 
free  end  of  the  heHcal  spring  of  problem  169  from  its  fixed  end ; 
the  initial  stretch  of  the  spring  being  f  of  an  inch. 

171.  Plot  a  curve  of  which  the  abscissas  represent  elapsed  time 
and  of  which  the  ordinates  represent  the  varying  distance  of  the 
middle  point  of  the  helical  spring  of  problem  169  from  its  fixed 
end  ;  the  initial  stretch  being  |  of  an  inch. 

172.  A  steel  rod  6  feet  long  is  supported  at  its  center,  as  shown 
in  Fig.  172/,  and  one  end  of  the  rod  is  stroked  with  a  rosined 


cloth,  causing  the  rod  to  vibrate  lengthwise.     Required  the  num- 
ber of  vibrations  made  by  the  rod  per  second. 

Note. — When  a  rod  is  supported  as  indicated  in  the  figure,  each  end  of  it  oscillates 
in  the  manner  represented  in  Figs.  155  to  158.  The  density  of  the  steel  is  7.78 
times  62  J  pounds  per  cubic  foot  and  its  stretch  modulus  is  30,cx)0,ooo  pounds  per 
square  inch. 

173.  A  brass  rod  11.63  ^^^^  long  is  mounted  as  indicated  in 
Fig.  1 7  2/,  and  when  stroked  with  a  rosined  cloth  it  gives  a  musical 
tone  in  unison  with  a  standard  tuning  fork  which  has  a  frequency 
of  512  complete  vibrations  per  second.     Calculate  the  stretch 


2^%'  ELEMENTS   OF   MECHANICS. 

modulus  of  brass,  its  density  being  8.4  times  62  J  pounds  per  cubic 
foot. 

174.  Assuming  that  the  face  of  a  hammer  comes  squarely 
against  the  end  of  a  steel  rod,  calculate  the  greatest  velocity  of 
the  hammer  which  will  not  batter  the  end  of  the  rod.  The 
resilience  of  the  steel  of  the  rod  is  8,640  foot-pounds  per  cubic 
foot. 

Note. — An  endwise  hammer  blow  on  a  steel  rod  produces  a  wave  of  compression 
of  which  the  velocity-phase  is  equal  to  the  velocity  of  the  hammer,  as  explained  in 
Art.  136.  The  compression  which  exists  in  this  wave  is  a  compression  of  which  the 
potential  energy  per  unit  volume  is  equal  to  the  kinetic  energy  per  unit  volume  due  to 
the  motion  of  the  steel.  The  battering  of  the  end  of  the  rod  is  due  to  the  compres- 
sion of  the  steel  beyond  the  elastic  limit. 

175.  The  velocity  of  an  air  wave  along  an  air  pipe  is  1,000 
feet  per  second.  The  air  in  the  pipe  is  slightly  compressed,  and 
then  one  end  of  the  pipe  is  suddenly  opened.  The  pipe  is  25  feet 
long.  How  many  complete  oscillations  does  the  air  in  the  pipe 
make  per  second  ? 

Note.  An  air  wave  is  reflected  from  the  open  end  of  a  pipe  in  the  same  way  that 
a  ware  of  compression  or  stretch  is  reflected  from  the  free  end  of  a  steel  rod. 


INDEX. 


Acceleration,  angular,  definition  of,  128 

definition  of,  47,  68 

discussion  of,  47 

of  gravity,  determination  of,  141 
Active  and  inactive  forces,  107 
Adhesion,  21 1 
Aeolotropy,  170 
Air  waves,  266 
Amplitude,  definition  of,  88 
Angle,  measurement  of,  21 

units  of,  21 
Angular  acceleration,  definition  of,    128 

velocity,  definition  of,  128 
units  of,  128 
Archimedes'  principle,  206 
Area,  measurement  of,  23 

units  of,  23 
Atmospheric  pressure,  198 
Balance,  the  analytical,  25 
Balancing  of  machine  parts,  98 

Ball-bat,  the  problem  of  the,  146 
Barometer,  the,  203 
Base-ball,  curved  pitching  of,  78 
Beam,  discussion  of  the  bent,  176 
Beaume's  hydrometers,  211 
Belts  on  pulleys,  behavior  of,  82 
Bourdon,  gauge,  the,  205 
Boyle's  law,  186 
Bulk  modulus,  185 
Buoyancy,  center  of,  206 
Buoyant  force  of  fluids,  2c6 

C.  g.  s.  system  of  units,  35 

Caisson,  the,  214 

Calculus,  differential,  method  of,  43 

integral,  method  of,  43 
Canal,  waves  in  a,  254 
Capillary  phenomena,  211 
Center  of  buoyancy,  206 


Center  of  gravity  of  a  body,  58 

of  mass  of  a  body,  58,  67 
equations  of,  98 
motion  of,  96 
Centrifugal  drier,  82 

force,  81 
Chladni's  figures,  275 
Chronograph,  the,  29 
Circular  translatory  motion,  79 
Clock,  the,  29 
Closed  system,  92 
Cohesion,  2H 
Component  of  a  vector,  40 
Compressibility  of  gases,  186 

coefficient  of,  186 
Conservation  of  energy,  119 

of  momentum,  94 
Constant  and  variable  quantities,  41 
Continuity,  principle  of,  42 
Corioli's  law,  151 
Correspondence  between  translatory  and 

rotatory  motion,  137 
Couple,  definition  of  a,  57 
Curves,  railway,  discussion  of,  83 
Cycle,  definition  of,  87 
Cyclones,  224 

D'Alembert's  principle,  59 
Density,  definition  of,  27 
measurement  of,  27 
Derived  and  fundamental  units,  34 
Differential  calculus,  method  of,  43 
Dimensions  of  units,  36 
Discharge  rate  of  a  stream,  225 
Dispersion  of  light,  basis  of,  250 
Displacement,  definition  of,  68 
Dividing  engine,  the,  19 
Drier,  the  centrifugal,  82 
Dynamics,  65 
Dyne,  definition  of  the,  7 1 


279 


28o 


INDEX. 


Eddy  friction  of  fluids,  239 

of  ships,  242 
Efficiency,  definition  of,    112 
Efflux  of  a  liquid  from  a  tank,  229 
Elastic  fatigue,  184 

hysteresis,  183 

lag,  183 

limit,  169,  181 
Elasticity,    163 
Energy,  conservation  of,   1 19 

definition  of,  113 

kinetic,   114,  115 

potential,  114,  116 
Equilibrium,  conditions  of,  55,  56 

definition  of,  54 

of  floating  body,  207 

stable,  unstable  and  neutral,  207 
Erg,  the,  109 

Falling  bodies,  74 
Flame,  the  sensitive,  223 
Floating  body,  equilibrium  of,  207 
Flow,  lamellar,  220 

permanent  and  varying  states  of,  219 

rotational  and  irrotational,  221 

simple,  219 
Fluid  friction,  238 

the  ideal  frictionless,  227 
Fluids  and  solids,  definition  of,  168 
Foot-pound,  the,  109 
Force  and  its  effects,  65 

measurement  of,  72 

moment  of,  see  torque, 

-polygon,  the,  38 

units  of,    71 

types  of,  67 
Forces,  active  and  inactive,  107 
F.  p.  s.  system  of  units,  35 
Frequency,  definition  of,  87 
Friction,  105 

angle  of,  106 

coefficient  of,  106 

of  fluids,  238 
Front  of  wave,  273 
Fundamental  and  derived  units,  34 

Gas,  definition  of  a,  169 


Gauge  tester,  the,  206 

Gradient,  space,  definition  of,  46 

Graduate,  the,  24 

Gravity,  determination  of  the  accelera- 
tion of,  141 
pendulum,  the,  139 
table  of  accelerations  of,  143 

Gyration,  radius  of,  132 

Gyroscope,  the,  147 

Harmonic  motion,  87 

motion,  examples  of,  90 

rotatory  motion,  138 
Heaviside's,  theory  of  wave  pulses,   253 
Hodograph,  definition  of,  48 
Hook's  law  of  elasticity,  169 
Hoop,  the  rotating,  85 
Horse-power,  definition  of,  1 10 
Hydraulic  ram,  the,  222 
Hydraulics,  218 
Hydrometer,  the,  211 
Hydrostatic  press,  the,  199 

pressure,  184 
Hydrostatics,  198 
Hysteresis,  elastic,  183 

Impact,  95 

Impulse,  definition  of,  144 
Inertia,  definition  of,  70 
Integral,  calculus,  method  of,  43 
Isotropic  strain,  185 

potential  energy  of,  186 
Isotropy,  170 

Jet,  impact  of  a  water,  233 

pump,  the,  232 

reaction  of  a  water,  233 
Joule,  the,  109 

Kater's  reversion  pendulum,  142 
Kilogram,  definition  of,  26 
Kinematics  of  a  rigid  body,  155 
Kinetic  energy,  114,  115 

energy  of  a  liquid,  229 
Kinetics,  see  dynamics 

Legal  units,  36 
Length,  measurement  of,  18 
units  of,  1 8 


INDEX. 


281 


Light,  dispersion  of,  basis  of,  ^50 
Liquid,  definition  of  a,  169 

energy  of  a,  227 
Longitudinal  strain,  potential  energy  of, 

175 
stress  and  strain,  1 7 1 

Magdeburg  hemispheres,  the,  198 
Manometers,  204 
Marriotte's  law,  see  Boyle's  law 
Mass,  center  of,  see  center  of  mass 

definition  of,  14,  25 

measurement  of,  26 

units  of,  26 
Measurement,  12,  18 

of  angle,  21 

of  area,  23 

of  density,  27 

of  force,  72 

of  length,  18 

of  mass,  26 

of  power,  1 10 

of  time,  28 

of  volume,  24,  28 
Measure  of  physical  quantity,  definition 

of,  33 
Medium,  wave,  249 
Metacenter  of  a  floating  body,  210 
Meter,  definition  of,  18 
Modulus,  the  elastic,  see  stretch  modulus, 

bulk  modulus  and  slide  modulus. 
Molecular  theory,  the,  248 
Moment  of  a  force,  see  torque, 

of  inertia,  definition  of,  129,  133 
units  of,  132 
Moments  of  inertia  about  parallel  axes, 

134 
comparison  of,  138 
table  of,  133 
origin  of,  56 
Momentum,  conservation  of,  94 

definition  of,  94 
Motion,  laws  of,  69 
rotatory,  65 
the  laws  of,  3 
translatory,  65 
types  of,  65 


Newton's  laws  of  motion,  3, 
Neutral  equilibrium,  207 
Nodes,  definition  of,  274 


69 


Orbit,  definition  of,  48 
Oscillation,  simple  and  compound  modes 
of,  274 
of  a  steel  rod,  259 
Oscillations,  forced  and  free,  276 

Particle,  definition  of,  66 
Particles,  systems  of,  92 
Pascal's  principle,  185 
Pendulum,  the  gravity,  139 

Kater's  reversion,  142 

the  simple,  91 

the  torsion,  138 
Perpetual  motion,  118 
Phase  difference,  definition  of,  88 

reversal  by  reflection  of  waves,  263 
Physical  measurement,  12,  18 
Physics,  the  science,  definition  of,  15 
Pipe,  tension  in  the  walls  of  a,  199 
Pitot  tube,  the,  236 
Planimeter,  theory  of  the,  23 
Poggendorff's    method     for     measuring 

angle,  22 
Poisson's  ratio,  173 
Polygon,  the  force,  38 
Potential,  definition  of,  50 

energy,  114,  116 

of  isotropic  strain,  186 
of  a  liquid,  227 
of  longitudinal  strain,  175 
Pound,  definition  of,  26 
Poundal,  definition  of  the,  71 
Pound- weight,  definition  of,  71 
Power,  no 

measurement  of,  no 

units  of,  no 
Precession  of  the  earth's  axis,  152 
Precessional  rotation,  examples  of,  152- 

155 

rotatory  motion,  147 
Pressure,  atmospheric,  198 
gauges,  204 
due  to  gravity,  200 


282 


INDEX. 


Pressure,  hydrostatic,  184 

measurement  of,  203 
Principal  pulls  of  a  stress,  193 

stretches  of  a  strain,  193 
Principle  of  Archimedes,  206 

of  Pascal,  185 
Projectiles,  75 
Prony  brake,  the,  112 
Pulses,  wave,  249 
Pupin's  system  of  telephonic  transmission, 

253 
Radian,  definition  of,  21 
Radius  of  gyration,  definition  of,  132 
Railway  curves,  discussion  of,  83 
Ram,  the  hydraulic,  222 
Rate  of  change,  definition  of,  42 
Reflection  with  and  without  phase  rever- 
sal, 263 
Resilience,  182 
Resolution  of,  vectors,  40 
Resonance,  276 
Reversion  pendulum,  the,  142 
Rigid  body,  definition  of,  66 

kinematics  of,  155 
Rod,  discussion  of  a  twisted,  191 
Rolling  wheel,  equivalent  mass  of,  135 
Rotation,     precessional,     examples     of, 

152-155 
Rotatory  motion,  65 

precessional,    147 
and  translatory  motion  correspond- 
dence,  137 

Scalar  quantities,  37 

and  vector  products  and  quotients,  40 
Segments,  vibrating,  definition  of,  274 
Sensitive  flame,  the,  223 
Shearing  stress  and  strain,  187 
Ships,  resistance  of,  242 
Shock,  liquid,  definition  of,  245 
Skin  friction  of  ships,  242 
Slide  modulus,  190 
Slug,  definition  of,  see  preface. 
Smoke  rings,  223 

Solids  and  fluids,  definition  of,  168 
Sound  waves,  266 
Specific  gravity,  definition  of,  27 


Specific  gravity,  measurement  of,  28 
Spring  balance  or  scale,  the,  25 
Stable  equilibrium,  207 
Statics,  simple,  54 
Steel  rod,  waves  in,  259 
Strain  and  stress,  163 
axes  of,  193 

due  to  sudden  load,  269 
isotropic,  185 

potential  energy  of,  186 
longitudinal,  potential  energy  of,  175 
principal  stretches  of,  193 
Strains,    homogeneous    and    non-homo- 
geneous, 163 
Stream,  discharge  rate  of,  225 

lines,  219 
Streams,  gauging  of,  237 
Strength,  tensile,   181 
Stress  and  strain,  163 

general  equations  of,  193 
longitudinal,  171 
shearing,  187 
types  of,  1 7 1 
axes  of,  193 
on  a  section,  193 
principal  pulls  of,  193 
tangential,  193 
Stresses,    homogeneous   and    non-homo- 
geneous, 163 
Stretch  moduli,  table  of,  174 
modulus,  173 

determination  of,  174 
Strings,  vibration  of,  269 
Surface  tension  of  a  liquid,  212 
System,  closed,  92 
connected,  93 
Systems,  conservative,  117 
of  particles,  92 

Table  of  compressibilities,  186 
of  gravity  accelerations,  143 
of  stretch  moduli,  174 
of  moments  of  inertia,  133 
of  tensile  strengths,  182 

Tangential  stress,  193 

Telephone  lines,  loaded,  253,  257 

Tensile  strength,  i8l 


INDEX. 


283 


Time,  measurement  of,  23 

units  of,  28 
Tornadoes,  224 
Top,  the  spinning,  149 
Torque,  definition  of,  55 
Torricelli's  theorem,  230 
Torsion,  discussion  of,  191 

pendulum,  the,  138 
Translatory  motion,  65,  67 
in  a  circle,  79 

and    rotatory    motion,    correspond- 
ence between,  137 
Triangulation,  22 
Twisted  rod,  discussion  of,  191 

Unstable  equilibrium,  207 
Units  of  angle,  21 

of  angular  velocity,  128 

of  area,  23 

dimensions  of,  36 

of  force,  71 

fundamental  and  derived,  34 

legal,  36 

of  length,  18 

of  mass,  26 

of  moment  of  inertia,  132 

of  power,  no 

systems  of,  35 

of  time,  28 

of  volume,  24 

of  work,  109,  112 

Variable  and  constant  quantities,  41 
Vector  quantities,  37 

and  scalar  products  and  quotients,  40 
Vectors,  addition  of,  38 

variation  of,  47 
Velocity,  definition  of,  47,  68 
Venturi  water  meter,  the,  233 
Vernier,  the  1 8 

Vibration,  simple  and  compound  modes 
of,  274 

of  strings,  269 


Vibrations,  forced  and  free,  276 
Virtual  work,  principle  of,  120 
Viscosity,  183 

coefficient  of,  243 
Viscous  friction  of  fluids,  238-243 
Volume,  units  of,  24 

measurement  of,  24,  28 
Vortex  motion,  221 

rings,  223 

"Water  hammer,  the,  221 

waves  in  a  canal,  254 

waves,  complexity  of,  248 
Watt,  definition  of,   1 10 
Wave  of  arrest,  255 

of  starting,  255 

complete,  a,  256 

distortion,  253,  257 

friction  of  ships,  242 

front,  the,  273 

media,  249 

pulses  and  wave  trains,  249 

shape,  250 

theory,  the,  248 

application   to   oscillatory   mo- 
tion, 261 
Waves  in  air  and  water  pipes,  266 

in  a  canal,  254 

plane  and  spherical,  274 

pure  and  impure,  253,  257 

in  a  steel  rod,  259 

on  a  wire,  250,  268 
Weight,  definition  of,  25,  71 
Wheel,  equivalent  mass  of  a  rolling,  135 
Wire  waves,  250,  268 
Wires,  vibration  of,  269 
Work,  definition  of,  109 

units  of,  109 

virtual,  principle  of,  I20 

Yard,  definition  of,  1 8 

Yield  point  of  steel,  181 

Young' s  modulus,  see  stretch  modulus. 


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